
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A token is placed in one square of a  $m\times n$  board, and is moved according to the following rules:



- In each turn, the token can be moved to a square sharing a side with the one currently occupied.
- The token cannot be placed in a square that has already been occupied.
- Any two consecutive moves cannot have the same direction.


The game ends when the token cannot be moved. Determine the values of  $m$  and  $n$  for which, by placing the token in some square, all the squares of the board will have been occupied in the end of the game.

Present the answer in LaTex format: \boxed{Your answer}
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