Human: Standardize and paraphrase the full text and put it in <text> tags. You should follow these rules:
* Stick with the exact structure of the original text (e.g. don't add a plan if the text does not have a plan)
* Don't add steps no already present in the original text
* Don't remove steps present in the original text and don't shorten them
* Don't switch the order of steps in the original text
* Don't remove dead ends, tedious steps, or useless steps
* When a step contains lots of details, KEEP ALL THE DETAILS. You can change the format but you should NOT lose any details or intermediate computation / steps.
* Use English
* Use latex
* When making a plan, use “planning: + an enumerated list”
* When using a step of the plan, say “step x: copy the bullet of the plan”
* When manipulating equations, use statements of the form “doing X:” between each equation”
* When recalling an empirical quantity from memory, use the format “the value of X is about Y”
* When changing course, use a simple sentence like “this is wrong” or “this doesn't help with X” followed by a simple sentence saying what you will do next (e.g. “let's try something different”)
* Use “doing X” when announcing an action and “Let's try doing X” when trying an approach that may not work.
The result should be **roughly as long** as the original text. It's NOT a problem if you answer is very very long. DON'T shorten the text, don't use any ellipse, I need the full text. If this means having a very long output that is longer than the generation limit, then so be it.
<text>
To solve this equation, we can try to manipulate it into a more manageable form. Let's recall some useful trigonometric identities:
1. $\sin^2(x) + \cos^2(x) = 1$
Since both $\sin(x)$ and $\cos(x)$ in the original equation are raised to odd powers, it might be helpful to try to reduce these powers.
We are given:
$\sin^{11}(x) + \cos^{11}(x) = 1$
Let's multiply both sides by $\sin^2(x) + \cos^2(x)$ = 1. This gives:
$(\sin^{11}(x) + \cos^{11}(x))(\sin^2(x) + \cos^2(x)) = 1 \cdot (\sin^2(x) + \cos^2(x))$
Expanding the left-hand side:
$\sin^{13}(x) + \cos^2(x) \sin^{11}(x) + \sin^2(x) \cos^{11}(x) + \cos^{13}(x) = \sin^2(x) + \cos^2(x)$
We can rearrange this as:
$\sin^{13}(x) + \cos^{13}(x) = \sin^2(x) + \cos^2(x) - \cos^2(x) \sin^{11}(x) - \sin^2(x) \cos^{11}(x)$
$\sin^{13}(x) + \cos^{13}(x) = \sin^2(x)(1 - \cos^{11}(x)) + \cos^2(x)(1 - \sin^{11}(x))$
Hmm, this isn't reducing the powers as I had hoped. Let me try another approach.
I can use the identity $\sin^2(x) + \cos^2(x) = 1$ to rewrite the original equation:
$\sin^{11}(x) + \cos^{11}(x) = (\sin^2(x) + \cos^2(x))^5 \cdot (\sin(x) + \cos(x))$
Actually no, that's wrong. Sorry. Let's start over.
</text>
Standardize and paraphrase the full text and put it in <text> tags. You should follow these rules:
* Stick with the exact structure of the original text (e.g. don't add a plan if the text does not have a plan)
* Don't add steps no already present in the original text
* Don't remove steps present in the original text and don't shorten them
* Don't switch the order of steps in the original text
* Don't remove dead ends, tedious steps, or useless steps
* When a step contains lots of details, KEEP ALL THE DETAILS. You can change the format but you should NOT lose any details or intermediate computation / steps.
* Use English
* Use latex
* When making a plan, use “planning: + an enumerated list”
* When using a step of the plan, say “step x: copy the bullet of the plan”
* When manipulating equations, use statements of the form “doing X:” between each equation”
* When recalling an empirical quantity from memory, use the format “the value of X is about Y”
* When changing course, use a simple sentence like “this is wrong” or “this doesn't help with X” followed by a simple sentence saying what you will do next (e.g. “let's try something different”)
* Use “doing X” when announcing an action and “Let's try doing X” when trying an approach that may not work.
The result should be **roughly as long** as the original text. It's NOT a problem if you answer is very very long. DON'T shorten the text, don't use any ellipse, I need the full text. If this means having a very long output that is longer than the generation limit, then so be it.

Assistant: Understood. I will paraphrase and standardize the text according to the rule without shortening it. <text>
Let's manipulate the equation.

Recalling relevant trigonometric identities and facts.
1. $\sin^2(x) + \cos^2(x) = 1$
2. $\sin(x)$ and $\cos(x)$ in the original equation are raised to odd powers.

Let's try reducing these powers.
Starting with:
$\sin^{11}(x) + \cos^{11}(x) = 1$
Multiplying both sides by $\sin^2(x) + \cos^2(x)$ = 1:
$(\sin^{11}(x) + \cos^{11}(x))(\sin^2(x) + \cos^2(x)) = 1 \cdot (\sin^2(x) + \cos^2(x))$
Expanding the left-hand side:
$\sin^{13}(x) + \cos^2(x) \sin^{11}(x) + \sin^2(x) \cos^{11}(x) + \cos^{13}(x) = \sin^2(x) + \cos^2(x)$
Rearranging:
$\sin^{13}(x) + \cos^{13}(x) = \sin^2(x) + \cos^2(x) - \cos^2(x) \sin^{11}(x) - \sin^2(x) \cos^{11}(x)$
Rearranging:
$\sin^{13}(x) + \cos^{13}(x) = \sin^2(x)(1 - \cos^{11}(x)) + \cos^2(x)(1 - \sin^{11}(x))$
This doesn't help reducing these powers. Let's try something different.

Let's try using $\sin^2(x) + \cos^2(x) = 1$ to rewrite the equation.
$\sin^{11}(x) + \cos^{11}(x) = (\sin^2(x) + \cos^2(x))^5 \cdot (\sin(x) + \cos(x))$
This is wrong. Let's try something different.
</text>

Human: Standardize and paraphrase the full text and put it in <text> tags. You should follow these rules:
* Stick with the exact structure of the original text (e.g. don't add a plan if the text does not have a plan)
* Don't add steps no already present in the original text
* Don't remove steps present in the original text and don't shorten them
* Don't switch the order of steps in the original text
* Don't remove dead ends, tedious steps, or useless steps
* When a step contains lots of details, KEEP ALL THE DETAILS. You can change the format but you should NOT lose any details or intermediate computation / steps.
* Use English
* Use latex
* When making a plan, use “planning: + an enumerated list”
* When using a step of the plan, say “step x: copy the bullet of the plan”
* When manipulating equations, use statements of the form “doing X:” between each equation”
* When recalling an empirical quantity from memory, use the format “the value of X is about Y”
* When changing course, use a simple sentence like “this is wrong” or “this doesn't help with X” followed by a simple sentence saying what you will do next (e.g. “let's try something different”)
* Use “doing X” when announcing an action and “Let's try doing X” when trying an approach that may not work.
The result should be **roughly as long** as the original text. It's NOT a problem if you answer is very very long. DON'T shorten the text, don't use any ellipse, I need the full text. If this means having a very long output that is longer than the generation limit, then so be it.
<text>
This question is asking if a balloon made out of lead (Pb), rather than a typical material like latex or mylar, could float if it was filled with helium.
To determine if a lead balloon filled with helium could float, I'll need to:
1. Calculate the density of helium gas at room temperature
2. Calculate the required volume of helium for buoyancy
3. Given the density of lead, determine if it's physically possible to construct a lead balloon with a thickness that would allow the combined balloon and helium to be buoyant in air.
Let's start with the density of helium. Under standard conditions (room temperature and atmospheric pressure), helium has a density of about 0.1786 kg/m³.
Air, under the same conditions, has a density of about 1.225 kg/m³.
For a balloon to float, the combined weight of the balloon material and the helium inside must be less than the weight of the air displaced by the balloon.
The density of lead is about 11,340 kg/m³. Even if we could make an incredibly thin lead balloon, the weight of the lead would be enormous compared to the lifting capacity of the helium.
Let's imagine we make a spherical balloon with a radius of 1 meter. The volume of such a sphere is 4/3 x π x r³, which for r = 1 is about 4.19 m³.
</text>
Standardize and paraphrase the full text and put it in <text> tags. You should follow these rules:
* Stick with the exact structure of the original text (e.g. don't add a plan if the text does not have a plan)
* Don't add steps no already present in the original text
* Don't remove steps present in the original text and don't shorten them
* Don't switch the order of steps in the original text
* Don't remove dead ends, tedious steps, or useless steps
* When a step contains lots of details, KEEP ALL THE DETAILS. You can change the format but you should NOT lose any details or intermediate computation / steps.
* Use English
* Use latex
* When making a plan, use “planning: + an enumerated list”
* When using a step of the plan, say “step x: copy the bullet of the plan”
* When manipulating equations, use statements of the form “doing X:” between each equation”
* When recalling an empirical quantity from memory, use the format “the value of X is about Y”
* When changing course, use a simple sentence like “this is wrong” or “this doesn't help with X” followed by a simple sentence saying what you will do next (e.g. “let's try something different”)
* Use “doing X” when announcing an action and “Let's try doing X” when trying an approach that may not work.
The result should be **roughly as long** as the original text. It's NOT a problem if you answer is very very long. DON'T shorten the text, don't use any ellipse, I need the full text. If this means having a very long output that is longer than the generation limit, then so be it.

Assistant: Understood. I will paraphrase and standardize the text according to the rule without shortening it. <text>
Rephrasing the question:
Can a balloon made out of lead (Pb) float if it was filled with helium.

Planning:
1. Calculate the density of helium gas at room temperature
2. Calculate the required volume of helium for buoyancy
3. Given the density of lead, determine if it's physically possible

Step 1: Calculate the density of helium gas at room temperature
Under standard conditions (room temperature and atmospheric pressure), helium has a density of about 0.1786 kg/m³

Step 2: Calculate the required volume of helium for buoyancy
Under standard conditions (room temperature and atmospheric pressure), air has a density of about 1.225 kg/m³.
For a balloon to float, the combined weight of the balloon material and the helium inside must be less than the weight of the air displaced by the balloon.
Estimating the weight of the balloon material:
Lead has a density of about 11,340 kg/m³.
Observation: This is very large compared to what helium can usually lift. The lead balloon would have to be very thin.
Estimating the volume of the balloon:
Simplifying assumption: the balloon is a sphere with a radius $r$ of 1 meter.
The volume of a sphere is $4/3 \times \pi \times r^3$
Numerical computation: $4/3 \times \pi \times r^3 = 4/3 \times \pi \times 1^3 \approx 4.19$ The volume of the balloon is about $4.19 m^3$
</text>

Human: Standardize and paraphrase the full text and put it in <text> tags. You should follow these rules:
* Stick with the exact structure of the original text (e.g. don't add a plan if the text does not have a plan)
* Don't add steps no already present in the original text
* Don't remove steps present in the original text and don't shorten them
* Don't switch the order of steps in the original text
* Don't remove dead ends, tedious steps, or useless steps
* When a step contains lots of details, KEEP ALL THE DETAILS. You can change the format but you should NOT lose any details or intermediate computation / steps.
* Use English
* Use latex
* When making a plan, use “planning: + an enumerated list”
* When using a step of the plan, say “step x: copy the bullet of the plan”
* When manipulating equations, use statements of the form “doing X:” between each equation”
* When recalling an empirical quantity from memory, use the format “the value of X is about Y”
* When changing course, use a simple sentence like “this is wrong” or “this doesn't help with X” followed by a simple sentence saying what you will do next (e.g. “let's try something different”)
* Use “doing X” when announcing an action and “Let's try doing X” when trying an approach that may not work.
The result should be **roughly as long** as the original text. It's NOT a problem if you answer is very very long. DON'T shorten the text, don't use any ellipse, I need the full text. If this means having a very long output that is longer than the generation limit, then so be it.
<text>
Let's find the prime number smaller than 1000 whose sum of digit is equal to 10 by brute force:
2 is prime and the sum of its digits is 2 = 2, {x} != 10 so we skip it.
3 is prime and the sum of its digits is 3 = 3, {x} != 10 so we skip it.
5 is prime and the sum of its digits is 5 = 5, {x} != 10 so we skip it.
7 is prime and the sum of its digits is 7 = 7, we can ignore it since {x} is not equal to 10.
11 is prime and the sum of its digits is 1+1 = 2, but {x} is not equal to 10.
13 is prime and the sum of its digits is 1+3 = 4, we can ignore it since {x} is not equal to 10.
17 is prime and the sum of its digits is 1+7 = 8, we can ignore it since {x} is not equal to 10.
19 is prime and the sum of its digits is 1+9 = 10, this is a solution
23 is prime and the sum of its digits is 2+3 = 5, but {x} is not equal to 10.
29 is prime and the sum of its digits is 2+9 = 11, we can ignore it since {x} is not equal to 10.
31 is prime and the sum of its digits is 3+1 = 4, we can ignore it since {x} is not equal to 10.
37 is prime and the sum of its digits is 3+7 = 10, this prime works
41 is prime and the sum of its digits is 4+1 = 5, {x} != 10 so we skip it.
43 is prime and the sum of its digits is 4+3 = 7, we can ignore it since {x} is not equal to 10.
47 is prime and the sum of its digits is 4+7 = 11, {x} != 10 so we skip it.
53 is prime and the sum of its digits is 5+3 = 8, we can ignore it since {x} is not equal to 10.
59 is prime and the sum of its digits is 5+9 = 14, {x} != 10 so we skip it.
61 is prime and the sum of its digits is 6+1 = 7, {x} != 10 so we skip it.
67 is prime and the sum of its digits is 6+7 = 13, we can ignore it since {x} is not equal to 10.
71 is prime and the sum of its digits is 7+1 = 8, but {x} is not equal to 10.
73 is prime and the sum of its digits is 7+3 = 10, this prime works
79 is prime and the sum of its digits is 7+9 = 16, we can ignore it since {x} is not equal to 10.
83 is prime and the sum of its digits is 8+3 = 11, {x} != 10 so we skip it.
89 is prime and the sum of its digits is 8+9 = 17, {x} != 10 so we skip it.
97 is prime and the sum of its digits is 9+7 = 16, but {x} is not equal to 10.
101 is prime and the sum of its digits is 1+0+1 = 2, we can ignore it since {x} is not equal to 10.
103 is prime and the sum of its digits is 1+0+3 = 4, {x} != 10 so we skip it.
107 is prime and the sum of its digits is 1+0+7 = 8, {x} != 10 so we skip it.
109 is prime and the sum of its digits is 1+0+9 = 10, this is a solution
113 is prime and the sum of its digits is 1+1+3 = 5, we can ignore it since {x} is not equal to 10.
127 is prime and the sum of its digits is 1+2+7 = 10, we found a solution
131 is prime and the sum of its digits is 1+3+1 = 5, but {x} is not equal to 10.
137 is prime and the sum of its digits is 1+3+7 = 11, {x} != 10 so we skip it.
139 is prime and the sum of its digits is 1+3+9 = 13, we can ignore it since {x} is not equal to 10.
149 is prime and the sum of its digits is 1+4+9 = 14, we can ignore it since {x} is not equal to 10.
151 is prime and the sum of its digits is 1+5+1 = 7, but {x} is not equal to 10.
157 is prime and the sum of its digits is 1+5+7 = 13, {x} != 10 so we skip it.
163 is prime and the sum of its digits is 1+6+3 = 10, this is a solution
167 is prime and the sum of its digits is 1+6+7 = 14, but {x} is not equal to 10.
173 is prime and the sum of its digits is 1+7+3 = 11, {x} != 10 so we skip it.
179 is prime and the sum of its digits is 1+7+9 = 17, we can ignore it since {x} is not equal to 10.
181 is prime and the sum of its digits is 1+8+1 = 10, we found a solution
191 is prime and the sum of its digits is 1+9+1 = 11, we can ignore it since {x} is not equal to 10.
193 is prime and the sum of its digits is 1+9+3 = 13, we can ignore it since {x} is not equal to 10.
197 is prime and the sum of its digits is 1+9+7 = 17, {x} != 10 so we skip it.
199 is prime and the sum of its digits is 1+9+9 = 19, {x} != 10 so we skip it.
211 is prime and the sum of its digits is 2+1+1 = 4, but {x} is not equal to 10.
223 is prime and the sum of its digits is 2+2+3 = 7, but {x} is not equal to 10.
227 is prime and the sum of its digits is 2+2+7 = 11, {x} != 10 so we skip it.
229 is prime and the sum of its digits is 2+2+9 = 13, we can ignore it since {x} is not equal to 10.
233 is prime and the sum of its digits is 2+3+3 = 8, we can ignore it since {x} is not equal to 10.
239 is prime and the sum of its digits is 2+3+9 = 14, {x} != 10 so we skip it.
241 is prime and the sum of its digits is 2+4+1 = 7, {x} != 10 so we skip it.
251 is prime and the sum of its digits is 2+5+1 = 8, we can ignore it since {x} is not equal to 10.
257 is prime and the sum of its digits is 2+5+7 = 14, but {x} is not equal to 10.
263 is prime and the sum of its digits is 2+6+3 = 11, but {x} is not equal to 10.
269 is prime and the sum of its digits is 2+6+9 = 17, {x} != 10 so we skip it.
271 is prime and the sum of its digits is 2+7+1 = 10, we found a solution
277 is prime and the sum of its digits is 2+7+7 = 16, but {x} is not equal to 10.
281 is prime and the sum of its digits is 2+8+1 = 11, {x} != 10 so we skip it.
283 is prime and the sum of its digits is 2+8+3 = 13, we can ignore it since {x} is not equal to 10.
293 is prime and the sum of its digits is 2+9+3 = 14, but {x} is not equal to 10.
307 is prime and the sum of its digits is 3+0+7 = 10, this is a solution
311 is prime and the sum of its digits is 3+1+1 = 5, but {x} is not equal to 10.
313 is prime and the sum of its digits is 3+1+3 = 7, but {x} is not equal to 10.
317 is prime and the sum of its digits is 3+1+7 = 11, we can ignore it since {x} is not equal to 10.
331 is prime and the sum of its digits is 3+3+1 = 7, but {x} is not equal to 10.
337 is prime and the sum of its digits is 3+3+7 = 13, {x} != 10 so we skip it.
347 is prime and the sum of its digits is 3+4+7 = 14, we can ignore it since {x} is not equal to 10.
349 is prime and the sum of its digits is 3+4+9 = 16, we can ignore it since {x} is not equal to 10.
353 is prime and the sum of its digits is 3+5+3 = 11, {x} != 10 so we skip it.
359 is prime and the sum of its digits is 3+5+9 = 17, {x} != 10 so we skip it.
367 is prime and the sum of its digits is 3+6+7 = 16, {x} != 10 so we skip it.
373 is prime and the sum of its digits is 3+7+3 = 13, we can ignore it since {x} is not equal to 10.
379 is prime and the sum of its digits is 3+7+9 = 19, {x} != 10 so we skip it.
383 is prime and the sum of its digits is 3+8+3 = 14, we can ignore it since {x} is not equal to 10.
389 is prime and the sum of its digits is 3+8+9 = 20, we can ignore it since {x} is not equal to 10.
397 is prime and the sum of its digits is 3+9+7 = 19, but {x} is not equal to 10.
401 is prime and the sum of its digits is 4+0+1 = 5, but {x} is not equal to 10.
409 is prime and the sum of its digits is 4+0+9 = 13, {x} != 10 so we skip it.
419 is prime and the sum of its digits is 4+1+9 = 14, {x} != 10 so we skip it.
421 is prime and the sum of its digits is 4+2+1 = 7, {x} != 10 so we skip it.
431 is prime and the sum of its digits is 4+3+1 = 8, {x} != 10 so we skip it.
433 is prime and the sum of its digits is 4+3+3 = 10, this is a solution
439 is prime and the sum of its digits is 4+3+9 = 16, {x} != 10 so we skip it.
443 is prime and the sum of its digits is 4+4+3 = 11, but {x} is not equal to 10.
449 is prime and the sum of its digits is 4+4+9 = 17, but {x} is not equal to 10.
457 is prime and the sum of its digits is 4+5+7 = 16, but {x} is not equal to 10.
461 is prime and the sum of its digits is 4+6+1 = 11, but {x} is not equal to 10.
463 is prime and the sum of its digits is 4+6+3 = 13, {x} != 10 so we skip it.
467 is prime and the sum of its digits is 4+6+7 = 17, we can ignore it since {x} is not equal to 10.
479 is prime and the sum of its digits is 4+7+9 = 20, {x} != 10 so we skip it.
487 is prime and the sum of its digits is 4+8+7 = 19, we can ignore it since {x} is not equal to 10.
491 is prime and the sum of its digits is 4+9+1 = 14, but {x} is not equal to 10.
499 is prime and the sum of its digits is 4+9+9 = 22, but {x} is not equal to 10.
503 is prime and the sum of its digits is 5+0+3 = 8, {x} != 10 so we skip it.
509 is prime and the sum of its digits is 5+0+9 = 14, {x} != 10 so we skip it.
521 is prime and the sum of its digits is 5+2+1 = 8, but {x} is not equal to 10.
523 is prime and the sum of its digits is 5+2+3 = 10, this prime works
541 is prime and the sum of its digits is 5+4+1 = 10, this is a solution
547 is prime and the sum of its digits is 5+4+7 = 16, we can ignore it since {x} is not equal to 10.
557 is prime and the sum of its digits is 5+5+7 = 17, but {x} is not equal to 10.
563 is prime and the sum of its digits is 5+6+3 = 14, {x} != 10 so we skip it.
569 is prime and the sum of its digits is 5+6+9 = 20, we can ignore it since {x} is not equal to 10.
571 is prime and the sum of its digits is 5+7+1 = 13, but {x} is not equal to 10.
577 is prime and the sum of its digits is 5+7+7 = 19, we can ignore it since {x} is not equal to 10.
587 is prime and the sum of its digits is 5+8+7 = 20, {x} != 10 so we skip it.
593 is prime and the sum of its digits is 5+9+3 = 17, {x} != 10 so we skip it.
599 is prime and the sum of its digits is 5+9+9 = 23, {x} != 10 so we skip it.
601 is prime and the sum of its digits is 6+0+1 = 7, but {x} is not equal to 10.
607 is prime and the sum of its digits is 6+0+7 = 13, {x} != 10 so we skip it.
613 is prime and the sum of its digits is 6+1+3 = 10, we found a solution
617 is prime and the sum of its digits is 6+1+7 = 14, but {x} is not equal to 10.
619 is prime and the sum of its digits is 6+1+9 = 16, {x} != 10 so we skip it.
631 is prime and the sum of its digits is 6+3+1 = 10, we found a solution
641 is prime and the sum of its digits is 6+4+1 = 11, but {x} is not equal to 10.
643 is prime and the sum of its digits is 6+4+3 = 13, {x} != 10 so we skip it.
647 is prime and the sum of its digits is 6+4+7 = 17, {x} != 10 so we skip it.
653 is prime and the sum of its digits is 6+5+3 = 14, we can ignore it since {x} is not equal to 10.
659 is prime and the sum of its digits is 6+5+9 = 20, {x} != 10 so we skip it.
661 is prime and the sum of its digits is 6+6+1 = 13, we can ignore it since {x} is not equal to 10.
673 is prime and the sum of its digits is 6+7+3 = 16, {x} != 10 so we skip it.
677 is prime and the sum of its digits is 6+7+7 = 20, we can ignore it since {x} is not equal to 10.
683 is prime and the sum of its digits is 6+8+3 = 17, {x} != 10 so we skip it.
691 is prime and the sum of its digits is 6+9+1 = 16, but {x} is not equal to 10.
701 is prime and the sum of its digits is 7+0+1 = 8, but {x} is not equal to 10.
709 is prime and the sum of its digits is 7+0+9 = 16, {x} != 10 so we skip it.
719 is prime and the sum of its digits is 7+1+9 = 17, we can ignore it since {x} is not equal to 10.
727 is prime and the sum of its digits is 7+2+7 = 16, we can ignore it since {x} is not equal to 10.
733 is prime and the sum of its digits is 7+3+3 = 13, {x} != 10 so we skip it.
739 is prime and the sum of its digits is 7+3+9 = 19, we can ignore it since {x} is not equal to 10.
743 is prime and the sum of its digits is 7+4+3 = 14, we can ignore it since {x} is not equal to 10.
751 is prime and the sum of its digits is 7+5+1 = 13, {x} != 10 so we skip it.
757 is prime and the sum of its digits is 7+5+7 = 19, we can ignore it since {x} is not equal to 10.
761 is prime and the sum of its digits is 7+6+1 = 14, but {x} is not equal to 10.
769 is prime and the sum of its digits is 7+6+9 = 22, but {x} is not equal to 10.
773 is prime and the sum of its digits is 7+7+3 = 17, but {x} is not equal to 10.
787 is prime and the sum of its digits is 7+8+7 = 22, but {x} is not equal to 10.
797 is prime and the sum of its digits is 7+9+7 = 23, we can ignore it since {x} is not equal to 10.
809 is prime and the sum of its digits is 8+0+9 = 17, we can ignore it since {x} is not equal to 10.
811 is prime and the sum of its digits is 8+1+1 = 10, we found a solution
821 is prime and the sum of its digits is 8+2+1 = 11, {x} != 10 so we skip it.
823 is prime and the sum of its digits is 8+2+3 = 13, we can ignore it since {x} is not equal to 10.
827 is prime and the sum of its digits is 8+2+7 = 17, but {x} is not equal to 10.
829 is prime and the sum of its digits is 8+2+9 = 19, we can ignore it since {x} is not equal to 10.
839 is prime and the sum of its digits is 8+3+9 = 20, {x} != 10 so we skip it.
853 is prime and the sum of its digits is 8+5+3 = 16, but {x} is not equal to 10.
857 is prime and the sum of its digits is 8+5+7 = 20, but {x} is not equal to 10.
859 is prime and the sum of its digits is 8+5+9 = 22, we can ignore it since {x} is not equal to 10.
863 is prime and the sum of its digits is 8+6+3 = 17, {x} != 10 so we skip it.
877 is prime and the sum of its digits is 8+7+7 = 22, but {x} is not equal to 10.
881 is prime and the sum of its digits is 8+8+1 = 17, but {x} is not equal to 10.
883 is prime and the sum of its digits is 8+8+3 = 19, we can ignore it since {x} is not equal to 10.
887 is prime and the sum of its digits is 8+8+7 = 23, but {x} is not equal to 10.
907 is prime and the sum of its digits is 9+0+7 = 16, we can ignore it since {x} is not equal to 10.
911 is prime and the sum of its digits is 9+1+1 = 11, we can ignore it since {x} is not equal to 10.
919 is prime and the sum of its digits is 9+1+9 = 19, but {x} is not equal to 10.
929 is prime and the sum of its digits is 9+2+9 = 20, we can ignore it since {x} is not equal to 10.
937 is prime and the sum of its digits is 9+3+7 = 19, {x} != 10 so we skip it.
941 is prime and the sum of its digits is 9+4+1 = 14, {x} != 10 so we skip it.
947 is prime and the sum of its digits is 9+4+7 = 20, we can ignore it since {x} is not equal to 10.
953 is prime and the sum of its digits is 9+5+3 = 17, {x} != 10 so we skip it.
967 is prime and the sum of its digits is 9+6+7 = 22, but {x} is not equal to 10.
971 is prime and the sum of its digits is 9+7+1 = 17, {x} != 10 so we skip it.
977 is prime and the sum of its digits is 9+7+7 = 23, we can ignore it since {x} is not equal to 10.
983 is prime and the sum of its digits is 9+8+3 = 20, we can ignore it since {x} is not equal to 10.
991 is prime and the sum of its digits is 9+9+1 = 19, {x} != 10 so we skip it.
997 is prime and the sum of its digits is 9+9+7 = 25, {x} != 10 so we skip it.

<answer>{19, 37, 73, 109, 127, 163, 181, 271, 307, 433, 523, 541, 613, 631, 811}</answer>
</text>
Standardize and paraphrase the full text and put it in <text> tags. You should follow these rules:
* Stick with the exact structure of the original text (e.g. don't add a plan if the text does not have a plan)
* Don't add steps no already present in the original text
* Don't remove steps present in the original text and don't shorten them
* Don't switch the order of steps in the original text
* Don't remove dead ends, tedious steps, or useless steps
* When a step contains lots of details, KEEP ALL THE DETAILS. You can change the format but you should NOT lose any details or intermediate computation / steps.
* Use English
* Use latex
* When making a plan, use “planning: + an enumerated list”
* When using a step of the plan, say “step x: copy the bullet of the plan”
* When manipulating equations, use statements of the form “doing X:” between each equation”
* When recalling an empirical quantity from memory, use the format “the value of X is about Y”
* When changing course, use a simple sentence like “this is wrong” or “this doesn't help with X” followed by a simple sentence saying what you will do next (e.g. “let's try something different”)
* Use “doing X” when announcing an action and “Let's try doing X” when trying an approach that may not work.
The result should be **roughly as long** as the original text. It's NOT a problem if you answer is very very long. DON'T shorten the text, don't use any ellipse, I need the full text. If this means having a very long output that is longer than the generation limit, then so be it.

Assistant: Understood. I will paraphrase and standardize the text according to the rule without shortening it. <text>
Let's find the prime number smaller than 1000 whose sum of digit is equal to 10.
Using a brute force approach:
2 is prime. The sum of its digits is 2 = 2,
3 is prime. The sum of its digits is 3 = 3,
5 is prime. The sum of its digits is 5 = 5,
7 is prime. The sum of its digits is 7 = 7,
11 is prime. The sum of its digits is 1+1 = 2,
13 is prime. The sum of its digits is 1+3 = 4,
17 is prime. The sum of its digits is 1+7 = 8,
19 is prime. The sum of its digits is 1+9 = 10, this is a solution
23 is prime. The sum of its digits is 2+3 = 5,
29 is prime. The sum of its digits is 2+9 = 11,
31 is prime. The sum of its digits is 3+1 = 4,
37 is prime. The sum of its digits is 3+7 = 10, this is a solution
41 is prime. The sum of its digits is 4+1 = 5,
43 is prime. The sum of its digits is 4+3 = 7,
47 is prime. The sum of its digits is 4+7 = 11,
53 is prime. The sum of its digits is 5+3 = 8,
59 is prime. The sum of its digits is 5+9 = 14,
61 is prime. The sum of its digits is 6+1 = 7,
67 is prime. The sum of its digits is 6+7 = 13,
71 is prime. The sum of its digits is 7+1 = 8,
73 is prime. The sum of its digits is 7+3 = 10, this is a solution
79 is prime. The sum of its digits is 7+9 = 16,
83 is prime. The sum of its digits is 8+3 = 11,
89 is prime. The sum of its digits is 8+9 = 17,
97 is prime. The sum of its digits is 9+7 = 16,
101 is prime. The sum of its digits is 1+0+1 = 2,
103 is prime. The sum of its digits is 1+0+3 = 4,
107 is prime. The sum of its digits is 1+0+7 = 8,
109 is prime. The sum of its digits is 1+0+9 = 10, this is a solution
113 is prime. The sum of its digits is 1+1+3 = 5,
127 is prime. The sum of its digits is 1+2+7 = 10, this is a solution
131 is prime. The sum of its digits is 1+3+1 = 5,
137 is prime. The sum of its digits is 1+3+7 = 11,
139 is prime. The sum of its digits is 1+3+9 = 13,
149 is prime. The sum of its digits is 1+4+9 = 14,
151 is prime. The sum of its digits is 1+5+1 = 7,
157 is prime. The sum of its digits is 1+5+7 = 13,
163 is prime. The sum of its digits is 1+6+3 = 10, this is a solution
167 is prime. The sum of its digits is 1+6+7 = 14,
173 is prime. The sum of its digits is 1+7+3 = 11,
179 is prime. The sum of its digits is 1+7+9 = 17,
181 is prime. The sum of its digits is 1+8+1 = 10, this is a solution
191 is prime. The sum of its digits is 1+9+1 = 11,
193 is prime. The sum of its digits is 1+9+3 = 13,
197 is prime. The sum of its digits is 1+9+7 = 17,
199 is prime. The sum of its digits is 1+9+9 = 19,
211 is prime. The sum of its digits is 2+1+1 = 4,
223 is prime. The sum of its digits is 2+2+3 = 7,
227 is prime. The sum of its digits is 2+2+7 = 11,
229 is prime. The sum of its digits is 2+2+9 = 13,
233 is prime. The sum of its digits is 2+3+3 = 8,
239 is prime. The sum of its digits is 2+3+9 = 14,
241 is prime. The sum of its digits is 2+4+1 = 7,
251 is prime. The sum of its digits is 2+5+1 = 8,
257 is prime. The sum of its digits is 2+5+7 = 14,
263 is prime. The sum of its digits is 2+6+3 = 11,
269 is prime. The sum of its digits is 2+6+9 = 17,
271 is prime. The sum of its digits is 2+7+1 = 10, this is a solution
277 is prime. The sum of its digits is 2+7+7 = 16,
281 is prime. The sum of its digits is 2+8+1 = 11,
283 is prime. The sum of its digits is 2+8+3 = 13,
293 is prime. The sum of its digits is 2+9+3 = 14,
307 is prime. The sum of its digits is 3+0+7 = 10, this is a solution
311 is prime. The sum of its digits is 3+1+1 = 5,
313 is prime. The sum of its digits is 3+1+3 = 7,
317 is prime. The sum of its digits is 3+1+7 = 11,
331 is prime. The sum of its digits is 3+3+1 = 7,
337 is prime. The sum of its digits is 3+3+7 = 13,
347 is prime. The sum of its digits is 3+4+7 = 14,
349 is prime. The sum of its digits is 3+4+9 = 16,
353 is prime. The sum of its digits is 3+5+3 = 11,
359 is prime. The sum of its digits is 3+5+9 = 17,
367 is prime. The sum of its digits is 3+6+7 = 16,
373 is prime. The sum of its digits is 3+7+3 = 13,
379 is prime. The sum of its digits is 3+7+9 = 19,
383 is prime. The sum of its digits is 3+8+3 = 14,
389 is prime. The sum of its digits is 3+8+9 = 20,
397 is prime. The sum of its digits is 3+9+7 = 19,
401 is prime. The sum of its digits is 4+0+1 = 5,
409 is prime. The sum of its digits is 4+0+9 = 13,
419 is prime. The sum of its digits is 4+1+9 = 14,
421 is prime. The sum of its digits is 4+2+1 = 7,
431 is prime. The sum of its digits is 4+3+1 = 8,
433 is prime. The sum of its digits is 4+3+3 = 10, this is a solution
439 is prime. The sum of its digits is 4+3+9 = 16,
443 is prime. The sum of its digits is 4+4+3 = 11,
449 is prime. The sum of its digits is 4+4+9 = 17,
457 is prime. The sum of its digits is 4+5+7 = 16,
461 is prime. The sum of its digits is 4+6+1 = 11,
463 is prime. The sum of its digits is 4+6+3 = 13,
467 is prime. The sum of its digits is 4+6+7 = 17,
479 is prime. The sum of its digits is 4+7+9 = 20,
487 is prime. The sum of its digits is 4+8+7 = 19,
491 is prime. The sum of its digits is 4+9+1 = 14,
499 is prime. The sum of its digits is 4+9+9 = 22,
503 is prime. The sum of its digits is 5+0+3 = 8,
509 is prime. The sum of its digits is 5+0+9 = 14,
521 is prime. The sum of its digits is 5+2+1 = 8,
523 is prime. The sum of its digits is 5+2+3 = 10, this is a solution
541 is prime. The sum of its digits is 5+4+1 = 10, this is a solution
547 is prime. The sum of its digits is 5+4+7 = 16,
557 is prime. The sum of its digits is 5+5+7 = 17,
563 is prime. The sum of its digits is 5+6+3 = 14,
569 is prime. The sum of its digits is 5+6+9 = 20,
571 is prime. The sum of its digits is 5+7+1 = 13,
577 is prime. The sum of its digits is 5+7+7 = 19,
587 is prime. The sum of its digits is 5+8+7 = 20,
593 is prime. The sum of its digits is 5+9+3 = 17,
599 is prime. The sum of its digits is 5+9+9 = 23,
601 is prime. The sum of its digits is 6+0+1 = 7,
607 is prime. The sum of its digits is 6+0+7 = 13,
613 is prime. The sum of its digits is 6+1+3 = 10, this is a solution
617 is prime. The sum of its digits is 6+1+7 = 14,
619 is prime. The sum of its digits is 6+1+9 = 16,
631 is prime. The sum of its digits is 6+3+1 = 10, this is a solution
641 is prime. The sum of its digits is 6+4+1 = 11,
643 is prime. The sum of its digits is 6+4+3 = 13,
647 is prime. The sum of its digits is 6+4+7 = 17,
653 is prime. The sum of its digits is 6+5+3 = 14,
659 is prime. The sum of its digits is 6+5+9 = 20,
661 is prime. The sum of its digits is 6+6+1 = 13,
673 is prime. The sum of its digits is 6+7+3 = 16,
677 is prime. The sum of its digits is 6+7+7 = 20,
683 is prime. The sum of its digits is 6+8+3 = 17,
691 is prime. The sum of its digits is 6+9+1 = 16,
701 is prime. The sum of its digits is 7+0+1 = 8,
709 is prime. The sum of its digits is 7+0+9 = 16,
719 is prime. The sum of its digits is 7+1+9 = 17,
727 is prime. The sum of its digits is 7+2+7 = 16,
733 is prime. The sum of its digits is 7+3+3 = 13,
739 is prime. The sum of its digits is 7+3+9 = 19,
743 is prime. The sum of its digits is 7+4+3 = 14,
751 is prime. The sum of its digits is 7+5+1 = 13,
757 is prime. The sum of its digits is 7+5+7 = 19,
761 is prime. The sum of its digits is 7+6+1 = 14,
769 is prime. The sum of its digits is 7+6+9 = 22,
773 is prime. The sum of its digits is 7+7+3 = 17,
787 is prime. The sum of its digits is 7+8+7 = 22,
797 is prime. The sum of its digits is 7+9+7 = 23,
809 is prime. The sum of its digits is 8+0+9 = 17,
811 is prime. The sum of its digits is 8+1+1 = 10, this is a solution
821 is prime. The sum of its digits is 8+2+1 = 11,
823 is prime. The sum of its digits is 8+2+3 = 13,
827 is prime. The sum of its digits is 8+2+7 = 17,
829 is prime. The sum of its digits is 8+2+9 = 19,
839 is prime. The sum of its digits is 8+3+9 = 20,
853 is prime. The sum of its digits is 8+5+3 = 16,
857 is prime. The sum of its digits is 8+5+7 = 20,
859 is prime. The sum of its digits is 8+5+9 = 22,
863 is prime. The sum of its digits is 8+6+3 = 17,
877 is prime. The sum of its digits is 8+7+7 = 22,
881 is prime. The sum of its digits is 8+8+1 = 17,
883 is prime. The sum of its digits is 8+8+3 = 19,
887 is prime. The sum of its digits is 8+8+7 = 23,
907 is prime. The sum of its digits is 9+0+7 = 16,
911 is prime. The sum of its digits is 9+1+1 = 11,
919 is prime. The sum of its digits is 9+1+9 = 19,
929 is prime. The sum of its digits is 9+2+9 = 20,
937 is prime. The sum of its digits is 9+3+7 = 19,
941 is prime. The sum of its digits is 9+4+1 = 14,
947 is prime. The sum of its digits is 9+4+7 = 20,
953 is prime. The sum of its digits is 9+5+3 = 17,
967 is prime. The sum of its digits is 9+6+7 = 22,
971 is prime. The sum of its digits is 9+7+1 = 17,
977 is prime. The sum of its digits is 9+7+7 = 23,
983 is prime. The sum of its digits is 9+8+3 = 20,
991 is prime. The sum of its digits is 9+9+1 = 19,
997 is prime. The sum of its digits is 9+9+7 = 25,

<answer>{19, 37, 73, 109, 127, 163, 181, 271, 307, 433, 523, 541, 613, 631, 811}</answer>
</text>

Human: Standardize and paraphrase the full text and put it in <text> tags. You should follow these rules:
* Stick with the exact structure of the original text (e.g. don't add a plan if the text does not have a plan)
* Don't add steps no already present in the original text
* Don't remove steps present in the original text and don't shorten them
* Don't switch the order of steps in the original text
* Don't remove dead ends, tedious steps, or useless steps
* When a step contains lots of details, KEEP ALL THE DETAILS. You can change the format but you should NOT lose any details or intermediate computation / steps.
* Use English
* Use latex
* When making a plan, use “planning: + an enumerated list”
* When using a step of the plan, say “step x: copy the bullet of the plan”
* When manipulating equations, use statements of the form “doing X:” between each equation”
* When recalling an empirical quantity from memory, use the format “the value of X is about Y”
* When changing course, use a simple sentence like “this is wrong” or “this doesn't help with X” followed by a simple sentence saying what you will do next (e.g. “let's try something different”)
* Use “doing X” when announcing an action and “Let's try doing X” when trying an approach that may not work.
The result should be **roughly as long** as the original text. It's NOT a problem if you answer is very very long. DON'T shorten the text, don't use any ellipse, I need the full text. If this means having a very long output that is longer than the generation limit, then so be it.
<text>
{text}
</text>
Standardize and paraphrase the full text and put it in <text> tags. You should follow these rules:
* Stick with the exact structure of the original text (e.g. don't add a plan if the text does not have a plan)
* Don't add steps no already present in the original text
* Don't remove steps present in the original text and don't shorten them
* Don't switch the order of steps in the original text
* Don't remove dead ends, tedious steps, or useless steps
* When a step contains lots of details, KEEP ALL THE DETAILS. You can change the format but you should NOT lose any details or intermediate computation / steps.
* Use English
* Use latex
* When making a plan, use “planning: + an enumerated list”
* When using a step of the plan, say “step x: copy the bullet of the plan”
* When manipulating equations, use statements of the form “doing X:” between each equation”
* When recalling an empirical quantity from memory, use the format “the value of X is about Y”
* When changing course, use a simple sentence like “this is wrong” or “this doesn't help with X” followed by a simple sentence saying what you will do next (e.g. “let's try something different”)
* Use “doing X” when announcing an action and “Let's try doing X” when trying an approach that may not work.
The result should be **roughly as long** as the original text. It's NOT a problem if you answer is very very long. DON'T shorten the text, don't use any ellipse, I need the full text. If this means having a very long output that is longer than the generation limit, then so be it.

Assistant: Understood. I will paraphrase and standardize the text according to the rule without shortening it. <text>