<think>
Alright, let me try to figure out how to solve this maze. So first, I need to parse the given maze structure and understand where each wall is placed, then figure out a path from the origin to the target.

Let me start by writing down each cell with its coordinates and walls. The maze is given row by row, so rows 0 to 4, each with columns 0-4. Let me parse each row step by step.

Starting with the first line (row 0):
- At 0-0, there's an up_down_left_wall. That means walls on top, bottom, and left? Wait, but how does that make sense since it's the corner. Hmm, maybe "up_down_left_wall" means walls in those directions? So this cell can't move up, down, or left. Since it's at 0-0 (the top-left corner), moving up or left isn't possible, but the down wall would mean it can't move down into row 1-0? Wait, actually walls are between cells. Maybe I need to think differently.

Wait, perhaps the walls indicate which directions from this cell are blocked. So for example, up_wall would mean that moving up (north) is blocked. Similarly, left_wall means moving west is blocked.

So for cell 0-0 (row 0 column 0):

up_down_left_wall: so walls in up, down, and left. Since it's the top row, moving up isn't possible, but the presence of an 'up' wall here might mean that you can't move up out of this cell, but since it's the top row, maybe that wall is redundant. Similarly, left_wall would block moving west (but there's nothing to the west here since it's column 0). The down_wall would block moving south (into row1-0).

Wait, but the cell's walls indicate which adjacent cells are blocked. For example, if a cell has an up_wall, then moving north from that cell is blocked.

Wait, perhaps each wall in the cell represents a wall on its side. So for example:

- up_wall: there's a wall above the cell, meaning you can't go north from here.
- down_wall: there's a wall below the cell, so you can't go south.
- left_wall: can't move west (left)
- right_wall: can't move east.

If a cell has multiple walls like up_down_left, then all those directions are blocked. So for 0-0 (row0, col0), up_down_left_wall would have walls in those directions. But since this is the top-left corner, moving up or left isn't possible normally. The down wall would block moving to row1-0, and the left wall is redundant here.

Wait but maybe the walls are on the cell's own sides. So for example, a down_wall would be a wall below this cell (so the cell below can't come up?), or does it block the current cell from moving down? I think it's blocking the movement from this cell in that direction. So if you're in 0-0, and it has a down_wall, then moving south (down) is blocked. So the next cell below (row1-0) can't be entered from here.

Wait, this is a bit confusing. Let me see another example. The origin is at 2-4, which is in row2 column4. Let's look there first because that's where we start.

Looking at the third row (row2):

<|2-0|><|down_left_wall|><|blank|>
Wait no, let me parse the third row correctly. The rows are given as:

Row2's line is:

<|2-0|><|down_left_wall|><|blank|>
Wait, the actual line given for row2:

The third row (row 2) is:
<|2-0|><|down_left_wall|><|blank|>
Wait, let me parse each cell properly. Let's take the third line of the MAZE:

The maze data for row 2 (third line):

<|2-0|><|down_left_wall|><|blank|>
Then comes 2-1, which is <|up_right_wall|><|blank|>
Then 2-2: <|down_left_wall|><|target|>, meaning the target is here.
Then 2-3: <|down_right_wall|><|blank|>
And finally, 2-4 is <|left_right_wall|><|origin|>.

Wait, so the origin is at 2-4. So starting point is (row2, column4). The cell at 2-4 has a left-right wall. So left and right walls are present here. Since it's column 4, moving east (right) isn't possible anyway. The left wall means cannot move west? Wait the cell has walls on left and right directions, so from origin (2-4), you can't move right (since it's column4) but also cannot move left? Wait, the walls are left and right. So you can't go west (left) or east? Wait, but for a cell's walls: left_wall would block movement to the west (so you can't go from 2-4 to 2-3?), and right_wall would block movement east, but since it's already at column4, that's out of bounds. So in this case, the origin is surrounded by walls except maybe south and north? Wait but also check what other walls are present. The cell 2-4 has left_right, so can you go up or down? Since those walls aren't there. So from origin (2-4), you can move north to row1-4 or south to 3-4, but need to check the walls in other cells.

Wait actually, for movement direction from the current cell: The walls on this cell block certain directions. So if I'm at 2-4, and that cell has left_wall and right_wall (since it's a "left_right_wall"), then I can't move west or east. But the current cell allows movement north or south? Since those walls aren't present here.

Wait, the cell's own walls: left and right are blocked. So to move north (up) from 2-4 would depend on the wall in this cell's up direction. But since it doesn't have an up_wall, so movement north is allowed? Wait but need to check the cell above (row1-4) and see if its down_wall is present. Hmm, this might be getting complicated because the walls are on both sides.

Alternatively, perhaps for movement between cells, you can move to an adjacent cell if either:

- From the current cell's perspective, there is no wall blocking that direction (like moving north requires this cell doesn't have an up_wall), and
- The adjacent cell's corresponding direction also allows it (i.e., the cell above doesn't have a down_wall, because that would block movement from below).

Wait, maybe each wall is present on the cell's side. So for example:

Moving north from a cell requires that this cell has no up_wall, and the target cell (north) has no down_wall. Because the wall between them must not exist.

Wait, actually, perhaps walls are one-way? Like if a cell has an up_wall, that means you can't go north from it. But the adjacent cell's down wall would be a separate consideration? Hmm, maybe I should treat each direction as the current cell's wall.

Alternatively, to simplify: for a given cell (r,c), you can move in direction "up" if the current cell does not have an up_wall. Similarly, moving down requires no down_wall in current cell, etc.

Wait, the problem description says: "Walls: <|no_wall|>, <|up_wall|>, etc." So perhaps each cell has walls on their own sides. For example, if a cell has an up_wall, then you can't move north from it. Similarly, down_wall blocks moving south, etc.

Therefore, to go from cell A (r,c) to B (r-1,c), you need that A has no up_wall. Because moving north requires the current cell doesn't block upward movement.

Therefore, to navigate:

From any position (r,c), possible moves are:

- Up: if current cell has no up_wall, and the target row (r-1, c) is within bounds.

- Down: if current cell has no down_wall, and (r+1,c) is within bounds.

- Left: current cell has no left_wall, and (r,c-1) is within bounds.

- Right: current cell has no right_wall, and (r,c+1) is within bounds.

Also, the target cell has a "target" tag, and origin has an "origin".

Our goal is to go from the origin (2-4) to the target, which according to row2's second cell: 2-2 has the target. So target is at (row2, column2).

Wait let me confirm:

Looking back:

Row 2-2: <|down_left_wall|><|target|>. So yes, the cell at (2-2) has a target. The origin is at 2-4.

So, the path must go from (2,4) to (2,2).

Let me first map out the maze step by step.

First, I need to create a grid with each cell's walls and check positions.

Let me list all cells in rows 0-4, columns 0-4:

Starting with row 0 (first line):

<|0-0|><|up_down_left_wall|> → so, walls: up, down, left. So, cannot move north (but it's top row), can't move south (down wall?), and can't move left (already at 0, so no effect). To move right from here: check if cell has left_wall? Wait to go right, the current cell's left wall doesn't block it. To move east (right) from 0-0, we need to check if the cell has a right_wall? Wait no: moving east requires that this cell doesn't have a right wall. Since the cell's walls are up, down, left. So it has no right_wall, so you can move east to 0-1.

Wait, the cell's walls are given as "up_down_left_wall", so it has up, down, and left walls. So the cell can move east (since right_wall isn't present here?), but also, to move south (down) into row1-0 would be blocked because of the down_wall here, so you can't move south.

Proceeding row by row:

Row0 cells:

0-0: up, down, left walls. Direction blocked: north (already top), south (blocked by the cell's own down_wall?), and west (can't go left as it's 0). So possible movements are east if no right wall. Since the cell doesn't have a right_wall (only up, down, left are mentioned), so east is allowed. So from 0-0 can move right to 0-1.

But let's proceed with the origin being at (2,4), so maybe I need to track from there first.

Starting point: origin is at 2-4. Let's look at that cell:

Cell (2,4): <|left_right_wall|>. So the walls are left and right. Therefore:

- can't move west (left) because of left_wall?
- can't move east (right) since it's already column4, and the wall is present here, but also out of bounds.

But to move west from 2-4 would require column3, but the left_wall here blocks movement left. So can't move west.

Movement options from origin (2-4):

Check directions:

Up: To check if current cell has up_wall. The cell's walls are left and right, so no up or down walls. So moving north is allowed (if possible).

Going to row1-4:

Check the cell at row1, column4: 1-4 has left_right_wall. Let's look back:

Row1-4: in the second line, after 1-3: <|up_left_right_wall|><|blank|> then comes 1-4.

Looking at row1: the line is:

<|1-0|><|up_left_wall|><|blank|><|1-1|><|down_right_wall|><|blank|><|1-2|><|left_right_wall|><|blank|><|1-3|><|up_left_right_wall|><|blank|><|1-4|><|left_right_wall|><|blank|>

So cell 1-4 has left and right walls. So the current cell (2-4) can move north to 1-4, but need to check the wall in the current cell for moving up. Since origin's cell (2-4) has no up_wall, so movement north is possible.

But then moving to 1-4: let's see the walls of 1-4. Since it has left_right_wall, so cannot move west or east from there. But to go north further, check if 1-4 has up_wall? The cell's walls are left and right. So it can move north (row0-4?) Wait row1-4's up direction would be to row0-4? Let me see.

Alternatively, moving north from 1-4 would require checking if 1-4 has an up_wall. Since it's walls are left and right, so no up_wall. Therefore can move north to row0-4?

Wait but that's possible, but let me check the cell at 0-4.

Looking back:

Row0 has cells up to 0-4. The line for row0 ends with <|0-4|><|up_right_wall|>.

Cell 0-4: up and right walls. Since it's the last column (4), moving east isn't possible. But up_wall blocks moving north, but it's already on row0, so no.

So from 1-4 moving north to 0-4 is allowed (if there's no blocking wall between them). Wait, the current cell 1-4 can move north since it has no up_wall. The target cell (0,4) has an up_wall? No, the 0-4's walls are up and right. The cell above (row-1) doesn't exist, so moving north from 0-4 is impossible.

But back to the path: starting at origin (2,4):

Option 1: Move north to row1-4.

From there, can we proceed further? Let's see:

At row1-4 (left_right_wall), so movement directions allowed are north, south, since left/right blocked.

Trying to move south: back to 2-4, which isn't helpful.

Moving north from row1-4 would go to 0-4, which is possible (since no up_wall in row1-4), but then moving further north isn't possible. So maybe that path is a dead end.

Alternatively, from origin's cell (2-4), can we move down to row3-4?

Checking the current cell's down_wall. Origin has walls left and right, so no down_wall.

So moving south to 3-4:

Looking at cell 3-4's walls: in row3, the line is:

<|3-0|><|up_left_right_wall|><|blank|> ...

Looking for 3-4: in the fourth row's data (row3):

<|3-0|><|up_left_right_wall|> ...

Then 3-1: <|down_left_wall|>,

3-2: <|up_down_wall|>,

3-3: <|up_right_wall|>,

3-4 has a left_right_wall? Let me parse row3 again:

The fourth line (row3):

<|3-0|><|up_left_right_wall|><|blank|>
<|3-1|><|down_left_wall|><|blank|>
<|3-2|><|up_down_wall|><|blank|>
<|3-3|><|up_right_wall|><|blank|>
<|3-4|><|left_right_wall|><|blank|>

So cell 3-4 has left and right walls. So moving south from origin (2-4 to 3-4):

The current cell can move down because it has no down_wall, and the target cell (3-4) must not have an up_wall. Since 3-4's walls are left and right, so it can accept movement from above (since no up_wall there).

Therefore, moving down to 3-4 is allowed. Let's see where that leads.

At 3-4, walls are left and right. So can move north (back to origin) or south? Wait 3-4's down direction would be row4-4. Let me check row4.

Looking at the fifth line (row4):

<|4-0|><|down_left_wall|>
Then 4-1: up_down,

Continuing to 4-4:

Looking at row4's last cell:

<|4-3|><|down_wall|><|blank|>
<|4-4|><|down_right_wall|><|blank|>

Wait, for 3-4 moving south to row4-4 (since column is same):

The cell at 4-4 has <|down_right_wall|>. So down and right walls. But since it's the bottom row, moving south isn't possible.

The cell 3-4 can move down to 4-4, but 4-4 has a down wall (which doesn't block since we're coming from above). Wait, the incoming to 4-4 would require that 3-4 has no down_wall (which it doesn't, since its walls are left and right), and the 4-4 has no up_wall (since cell's own walls are down and right. So yes, can move to 4-4.

But proceeding further from there would be stuck as bottom row. Not helpful for reaching target (2,2).

Hmm, perhaps moving from origin north to 1-4 first?

Wait let me see other options. From origin (2,4), another direction is moving west to column3:

Can we move left? The origin has a left_wall, so yes. Wait the cell's walls are left and right. So moving west (left) is blocked by its own left_wall.

So can't move west from 2-4, only possible directions north and south.

Thus, the initial moves are either N to 1-4 or S to 3-4.

Let me try first the path going north to row1-4, then see:

At 1-4: walls left and right. So can't move east or west, only north/south.

Moving north to 0-4:

At 0-4: walls up and right. Since it's the top row, moving north isn't possible. Can move south back to 1-4 or east? But right wall blocks that.

Alternatively, moving down from row0-4 to 1-4 again.

This path doesn't seem helpful towards the target (2,2). Let's try other options.

Alternatively, going down to 3-4 first.

From origin to 3-4, then south to 4-4, which might not help. So let's see another direction.

Wait perhaps the target is at (2,2). Let me see the path from origin to there.

Origin is at 2-4. To get to 2-2, we need to move west twice (columns 4→3→2) along row2.

But in the current cell 2-4, moving west is blocked by left_wall. So can't move directly to column3 from 2-4.

So must go around via other rows, perhaps going up or down to another row where movement west is possible.

Let me look at the target cell (2-2):

Cell 2-2 has a down_left_wall. So walls: down, left.

Thus, the target is at (2,2). To get here from origin, probably need to move west through row1 or 3.

Let me try moving south first: from origin (2-4) to row3-4, then perhaps move west?

But in 3-4: walls left and right. So can't go west (left_wall).

Hmm, perhaps going north first to row1-4, then east? But can't go west from there. Wait no.

Alternatively, maybe going up to row1-4, then moving south? Not helpful.

Another option: moving down from origin to 3-4, then down to 4-4, but that's probably a dead end.

Alternatively, from origin (2-4), moving north to row1-4, then west? But the cell 1-4 has left_right walls blocking left and right. Hmm.

Wait perhaps I need to go further east, but can't.

Alternatively, let's try a different path. Let me look at the maze structure again.

Starting from origin (2-4), can we go north to 1-4, then west? Let's see:

Row1-4 has left_right walls. So moving left (west) to 1-3:

Cell1-3 has up_left_right_wall.

Looking back at row1, cell1-3: the description was:

<|1-3|><|up_left_right_wall|>

So that cell has walls: up, left, and right. Therefore, moving west from 1-4 to 1-3 would be blocked by the right_wall of 1-3? Wait, actually to move left from 1-4 (which is column4), moving west would be to column3, which is cell1-3.

The cell 1-4 has a left wall (since it's left_right_wall), so you can't move west from 1-4 because of its own left_wall.

Therefore, moving west is blocked at row1-4.

Hmm, so perhaps need to try another direction.

Wait maybe instead of moving straight north or south from origin, but that's the only options.

Alternative idea: Let me try to map all possible moves step by step, maybe using a BFS approach mentally.

Starting point: (2,4). Need to get to (2,2).

Possible first moves:

1. North: to row1-4.

Let's explore that path:

Row1-4 (current):

From here, can go north to 0-4, which is a dead end, or down back to origin, or? Also check other directions. But since left and right are blocked here.

Wait, can I move west from row1-4? No because of left_wall.

Alternatively, going to 0-4 is the only new cell here. But that's a dead end.

So backtrack, trying moving south first:

Second option: Move down from origin to row3-4.

At row3-4 (left_right_wall), can move down to 4-4? Let's see:

Moving down from row3-4 (to 4-4):

Check if row3-4 has a down_wall? Its walls are left and right, so no.

Then row4-4 has a down_right_wall. Since we're moving south, the cell's up direction allows it (row4-4 has no up_wall?), so yes.

At 4-4, can move north back to row3-4, or perhaps east? But column5 is out of bounds.

Alternatively, moving left from row3-4: but blocked by the cell's left_wall (since it has left_right walls), so cannot move west.

Thus, this path also seems stuck.

Hmm, maybe I need to look for another route. Let's see the target location again: (2-2). To get there, perhaps moving west through row1?

Let me consider the target cell (2-2). Let's see what walls it has:

Cell 2-2: <|down_left_wall|>. So down and left walls.

So from the target, you cannot move south (down_wall), or west (left_wall). But can move north and east.

To reach the target, you need to come from above (row1-2), right (east from 2-3?), etc.

Wait let me check the cell to the east of target, which is 2-3:

Cell2-3 has <|down_right_wall|>. So walls down and right.

Thus, moving east from target to 2-3 is blocked by the left_wall of 2-3? Wait, no. To move east from target (2-2) to 2-3:

The current cell's right wall? The target has no right_wall (only down and left), so can move east to 2-3.

But the target cell (2-2) is the destination, so perhaps coming from above?

Alternatively, trying to approach via row1-2.

Let me see:

Row1 has cell1-2: <|left_right_wall|>.

Thus, moving west/east is blocked there. So row1-2 can only move north/south.

Hmm, this maze seems complex. Let me try to find a path step by step:

Starting at (2,4):

Option 1: Move North to row1-4. Let's see:

Path so far: north

Now at (1,4).

Possible moves here are south back, or north to (0,4), but let's try moving east? Wait no, can't move west or east due to left_right walls.

Wait, stuck here. Backtrack.

Option 2: Move down (S) to (3-4):

Path: S

Now at (3,4).

From here, can move south to (4-4):

Path: S, down again? Let's try moving South:

Path so far: S, down to (4-4).

Now at row4-4. Can't move further south, only back north to (3-4). Not useful.

Backtrack again.

Hmm, so maybe need to go east? But can't from origin.

Wait another thought: the target is in row2, so maybe need to move west through other rows.

Alternatively, from origin (2-4), can we move down first to 3-4, then left? But blocked by wall.

Alternatively, go north to row1-4, then move west is blocked but moving south? No.

Hmm. Maybe I need to look for other paths that loop around.

Wait, let's consider moving left from the origin isn't possible. So need to move north or south first, then turn west.

Suppose from origin:

Move South to (3-4), then move left? But blocked.

Alternatively, let's try going down to 3-4, then west isn't possible, so maybe move east? But can't.

Alternatively, let me look at another row:

Looking at row2-1 (cell to the west of target):

Cell 2-1 has up_right_wall.

So walls: up and right.

Therefore, moving east from there to 2-2 (the target) would require that the current cell's right wall is not present.

Wait, to move from 2-1 (west) to target's cell (east), the current cell has a right_wall, so blocked.

Hmm.

Wait maybe moving up to row1-2? Let me see:

Going from origin north to 1-4, then moving left blocked. Alternatively, another path.

Wait perhaps the solution requires going to row0 first and then looping around?

Alternatively, starting from origin (2-4):

Move south to row3-4, then move west isn't possible. Hmm.

Wait let's check the cell (2,3), which is adjacent to target:

Cell 2-3 has down_right_wall.

Thus, from (2-4) to (2-3) is blocked by left_wall of 2-4. Can't go west directly.

Hmm, this is tricky. Maybe I need to consider moving east? But origin's at column4, so that's the edge.

Wait another thought: perhaps moving north from origin to row1-4, then move west via some other route. Wait, but can't move west from there.

Alternatively:

Let me think of another path:

From origin (2-4), move South to row3-4. Then, moving west is blocked here, so can't go left.

But perhaps move down to row4-4, then west? But blocked by cell's walls.

Alternatively, moving north to row1-4, then west to some other cell?

Wait can't move west from there.

Hmm.

Perhaps I need to look at the entire maze structure more carefully, drawing a grid might help. Let me try to sketch mentally:

Rows 0-4, columns 0-4.

Origin at (2,4), target at (2,2).

Let me note all the walls for each cell:

Starting with row0:

Row0:
- 0,0: up_down_left → walls in those directions. Can move east (since no right wall?), can't go south (down_wall here blocks going to 1-0).

So from 0,0 can go east to 0,1.

Cell0,1 has up_right_wall. So blocks moving north (already top row), can move east? Let's see:

0,1 has right wall → so cannot go east. Can go south (down) if no down_wall here.

0,1's walls are up and right. So can move south (since no down_wall here), so to row1-1.

Similarly, proceeding:

But perhaps this is getting too long. Let me try to see the target cell (2,2):

To get here, perhaps coming from above (row1-2), which has left_right_wall. So row1-2 can't move east or west, but can go north/south.

Row1-2 has left_right walls → so cannot move east/west. Thus, to reach row1-2 must come from above (0-2) or below (row2-2).

Wait cell2-2 has a down wall, so cannot come from below. Thus must come from row0-2.

Cell0-2:

Looking back, in row0's cells:

Row0-2 has up_left_wall. So walls are up, left.

So can move south (no down wall?), so from 0-2 to 1-2.

Yes, because row0-2 has no down_wall (since it's "up_left_wall").

Thus, path could be:

0-2 →1-2. But 1-2 is blocked on left and right, so from there can only move north or south.

Going to 0-2 first requires getting to row0, which might be a long path.

Alternatively, another route:

Suppose I go from origin down to row3-4, then move west through row3? Let me check:

At row3-4: can't move west.

Row2, column 3 has down_right_wall → can I come from the north?

Let me see: to get to 2-3, you could move west from column4 (origin), but blocked.

Alternatively:

From origin, moving north to 1-4, then west is blocked, but maybe up again?

Wait row0-4 has up_wall → can't move north further.

Hmm.

Another idea: Let's try to see if moving left from the target is possible, and perhaps come from row1-2.

Wait to reach the target (2-2), you might have to come from row1-2 or row3-2, but row2-2 has down_left_wall:

Can come from above (row1-2), below (row3-2) or east?

Looking at row1-2:

Cell1-2 has left_right_wall → can't come from west/east. Must come from above or below.

To get to row1-2, need to go through row0-2 (since moving from below would hit the down_wall of target cell).

Row0-2 has up_left_wall.

Thus, to reach row1-2:

Need to move from 0-2 south to 1-2.

To get to row0-2, you can come from the west (row0-1) or east?

Wait row0-2's left wall blocks moving west, so must come from the right (east side)?

Row0-2 has left_wall blocking west movement, so coming from the east (row0-3?)

But row0-1 has up_right_wall, so can't go east past there?

Let me try to trace a path from the origin (2,4) all the way around to 0-0, but that might be too long.

Alternatively, let me try the following path:

Start at 2-4 (origin). Move down to 3-4.

Then, from 3-4 can't move west, so move down to 4-4.

Then go left along row4:

At row4-4, can move west to 4-3 (since column3 is west).

But row4-4 has a right_wall, so to move left (west) from 4-4 requires the current cell has a left wall? Wait:

Wait to move west (left) from 4-4, need that the current cell's left wall is not present. The walls of row4-4 are down and right, so the left wall isn't there. So can move west to 4-3.

At row4-3: which has down_wall.

So, moving from 4-4 to 4-3 is allowed (west).

Now at row4-3. Can move north, since it has down_wall?

Checking:

Row4-3's walls: <|down_wall|. So can move north.

Moving north to 3-3:

Cell 3-3 has up_right_wall.

Thus, can move north only if current cell (4-3) allows going up? Yes.

At 3-3: walls are up and right.

So, can move north (to row2-3?), but check the cell's up_wall.

Cell 3-3 has an up wall → so cannot move north from here back to row2-3.

Wait, moving north would require the current cell (3-3) has no up wall? But it does have an up_wall. So blocked.

Hmm, so can't move north from 3-3.

Alternatively, moving west (left) to row4-2:

At 3-3 can move left? Let's see cell 3-3 has a right_wall blocking going east, but can move west (left)?

To go from 3-3 to column2 (left), check cell's left wall.

The walls of 3-3 are up and right → no left_wall. So can move west to 3-2.

At row3-2: which has up_down_wall.

So, this cell can't move north or south. But can move east/west? Let's see:

Since it has up and down walls, so moving north or south is blocked.

Left and right directions are allowed unless blocked by cell's walls, which only up_down.

Thus, can move left (to 3-1) or right (to 3-3 again? No, already there)

Moving west to row3-1:

Row3-1 has down_left_wall.

So, can move north (since current cell has no up wall?), but to do that:

At 3-1, can move north (row2-1), since the cell has down_wall?

Wait row3-1's walls are down and left. So moving north requires that cell has no up_wall, so yes, can move to row2-1.

At row2-1: has up_right_wall.

So, walls are up and right. Can move south (back) or west?

Moving east from row2-1 to 2-2 is blocked by the right wall here.

Wait, to move east (right) from row2-1 to 2-2:

Current cell's right wall is present (since it has up_right_wall), so blocked.

Thus, can't proceed east to target.

Hmm, this path is getting too convoluted. Maybe another approach:

Let me try a different route starting from origin (2-4):

Move North to row1-4, then move West isn't possible.

Wait what if I move west first but can't, so must go down to row3-4, then move south to 4-4, then west to 4-3, west again to 4-2, etc.

Continuing that path:

From row4-3, can I move north to 3-3 (already did that).

Alternatively, moving west further:

From row4-2's cell: let me see.

Wait backtracking, after moving to 3-2 (up_down_wall):

From row3-2 can move left (west) to row3-1.

Row3-1 has down_left_wall (down and left walls). So can move north to row2-1.

At row2-1, has up_right_wall (up and right). Moving west would require going to 2-0?

Wait west from row2-1 is column0?

No, moving left (west) to 2-0. But let's see:

At row2-1, moving west (to column0):

The cell has up and right walls, so left direction isn't blocked.

Wait, to move west from 2-1 to 2-0:

The current cell (row2-1) has a left wall? Its walls are up and right. No, so can move west.

Thus, moving to row2-0? Wait column0:

Wait 2-1's west direction is 2-0.

Checking walls of row2-0:

Cell2-0 has down_left_wall → walls down and left. So from row2-1 moving west to 2-0:

The current cell (row2-0) has a left wall, but that would block movement from the east? Wait:

Wait to move west to row2-0, we need that the current cell (row2-1) has no left wall. Since it's walls are up and right, so yes.

Thus, moving west to row2-0 is allowed.

At 2-0: has down_left_wall → so cannot move south (row3-0) because of the down wall?

Wait row2-0's walls are down and left. So:

Can move north (since no up_wall), or east (to 2-1 again?), but also west isn't possible.

Moving north to row1-0:

Check if allowed.

Row2-0 can move north only if it has no up_wall. Since the walls are down and left → yes.

Thus, moving north to row1-0:

Row1-0 has up_left_wall.

So walls are up and left. Can move south back to 2-0, east to row1-1?

Moving east from row1-0 (to column1):

The current cell has left_wall, which blocks west, but moving east requires right?

Wait the cell's left_wall doesn't block moving east. To move east, check if row1-0 has a right wall?

The cell's walls are up and left. So can move east to row1-1 (column1).

At row1-1, has down_right_wall:

So walls are down and right.

Can move north (row0-1?), since no up_wall is present here.

 row1-1 can move north to 0-1, or south (blocked by down_wall), west?

West would be moving left to row1-0 again.

Going north to 0-1:

At row0-1, has up_right_wall (up and right walls).

Can move south back or east? Moving east would require checking row0-1's right wall → yes, so blocked.

Thus moving west to 0-0?

 row0-1 can move west (left) to 0-0, since no left wall there?

 row0-1's walls are up and right. So west movement is allowed (no left_wall here).

Thus moving to row0-0.

Now at 0-0, which has up_down_left_wall.

Can move east (since no right_wall?), yes.

 row0-0 can move east to 0-1, but already here.

Hmm, this is going in circles.

Perhaps another path:

At row2-1 (where we were earlier), can move north to row1-1?

Wait, let's see.

If I am at row2-1 (from moving west from 2-4 path), but this is getting too long. Maybe I need to consider that the target is two columns west of the origin, so perhaps there's a path through row1 or 3.

Wait let me try moving south from origin to row2-4, then down to row3-4, then west isn't possible.

Alternatively, move North again.

Wait origin can also move north to row1-4, then west isn't possible. But what if I go to the other direction?

Let me try this path step by step:

Start at (2-4). Move North to row1-4.

Then, from row1-4, can I move South again (back to origin) is not helpful.

Alternatively, moving north to row0-4:

(2,4 → 1,4 →0,4).

At row0-4, can move west (left) to row0-3?

 row0-4 has up_right_wall, so can move west.

 row0-3 has <|up_down_wall|>.

Wait looking at the first line:

Row0-4 is the last cell of row0. Let me check its walls: <|up_right_wall|>.

So, to move west from row0-4 (column4→3):

The current cell has a right wall, but that blocks east, not west.

Thus, moving west is allowed (no left_wall in row0-4? Its walls are up and right → yes, can move west to 0-3.

At row0-3: has up_down_wall → walls in both directions (up and down). So cannot move north or south.

Can move east back to 0-4, or west to row0-2.

Moving west to row0-2:

Since 0-3 has no left wall (its walls are up and down), so can move west to 0-2.

 row0-2 has up_left_wall.

At row0-2, can move south (down) since it doesn't have a down_wall.

 row0-2 has walls up and left, so moving south to 1-2.

 row1-2 has left_right_wall → cannot move east or west, but can move north/south.

From row1-2, moving north to 0-2 again or south to 2-2.

Wait, row1-2 can move south to 2-2?

Let me see:

 row1-2 has no down_wall (its walls are left and right), so yes.

Thus, moving south from row1-2 to target at 2-2 is possible!

Yes! So that path works:

Let me retrace the steps:

Starting at origin (2-4):

1. Move North to row1-4 → N

2. From 1-4, move west to 0-4 (wait no: can't move west from row1-4 because of its left_wall).

Wait, earlier I thought that to reach 0-4, you have to go from row1-4 to row0-4 by moving north.

Wait let's re-examine:

First move: N from 2-4 to row1-4.

Second move: North again to 0-4.

Then:

3. Move west from 0-4 to row0-3 (W)

 row0-4 has up_right_wall → can move west.

4. Move W again to 0-2 (W)

5. Then move south twice: S, S to reach row2-2.

Wait let's see step by step:

1. N from (2,4) to row1-4: N

At row1-4.

2. Move North (N) to row0-4: N

Now at 0,4.

3. Move West (W) to row0-3: W

At 0,3.

4. Move West (W) to row0-2: W

At 0,2.

5. Move South (S) to row1-2: S

At 1,2.

6. Move South again (S) to row2-2: S

Now at target!

Thus, the path is:

N N W W W S S ?

Wait let's count steps:

1. North to 1-4 (N)

2. North again to 0-4 (N) → total N steps: NN

3. West to row0-3 (W)

4. another west to 0-2 (another W → total W steps: WW after the first N)

5. another west? Wait from row0-2, can I proceed south?

Wait after step4 (at 0-2):

Move South to row1-2 (S)

Then another south to reach target at row2-2 (another S).

So total moves:

N, N, W, W, S, S.

Wait that's 6 moves. Let me check each step:

Start: (2-4)

1. N → row1-4 (allowed, since down_wall not present in 2-4?)

Yes, origin has no up_wall blocking moving north.

Wait the cell 2-4's walls are left and right, so N is allowed.

Moving to row1-4:

2. N again → 0-4 (allowed?)

Check if row1-4 has a down_wall? No, its walls are left and right.

So moving north from row1-4 (row1 to 0):

Yes, so now at 0-4.

3. W move west from row0-4 to 0-3:

Cell0-4 has walls up and right. Since moving west (left), the current cell's left wall isn't a concern.

Wait, to move west from column4→3 requires that row0-4 doesn't have a left_wall. Its walls are up and right, so yes.

Thus, moving west to 0-3.

Cell0-3 has up_down_wall.

4. Move west again to 0-2:

Cell0-3 has up_down walls: can move left?

Yes, since those are vertical walls. The left wall isn't mentioned here (the walls are up and down). So yes, moving west to 0-2 is okay.

At row0-2:

walls up and left.

5. Move south to row1-2 (S):

Check if 0-2 has down_wall? Its walls are up and left → no. So moving south to row1-2 is allowed.

At row1-2 (left_right walls):

Can move down to 2-2:

Yes, since row1-2 has no down_wall.

Thus, moving S (step5: S?), wait steps:

Wait after 0-2 → row1-2 is step5 (S)

Then another S to get to 2-2.

So total moves:

Step1: N

Step2: N → row0-4

Step3: W → 0-3

Step4: W →0-2

Step5:S→1-2

Step6:S→ target (2-2)

Thus the sequence is N, N, W, W, S, S.

So the moves are: N N W W S S

Let me confirm each step:

- Starting at 2-4:

1. North to row1-4: yes.

At row1-4 (left_right_wall):

2. North to 0-4: yes.

At row0-4 (up_right_wall):

3. West to 0-3: yes.

At row0-3 (up_down_wall):

4. West to 0-2: yes.

At row0-2 (up_left_wall):

5. South to 1-2: yes.

At row1-2 (left_right_wall):

6. South to 2-2: yes, because it can move down (no down_wall in row1-2), and 2-2 has no up_wall blocking?

Wait, target cell (2-2) has a down_left_wall. So it can be entered from above.

Thus, this path works.

Therefore the sequence is N N W W S S → but let me write it as individual steps:

<|up|> <|up|> <|left|> <|left|> <|down|> <|down|>

Yes, because moving north twice (up), then west (left) twice, and south (down) twice.

Thus the answer should be: up up left left down down
</think>

<|up|> <|up|> <|left|> <|left|> <|down|> <|down|>