
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
In a corner of a chessboard of size $m \times n$ squares, there is a rook. Two players take turns moving the rook either vertically or horizontally any number of squares; the rook is not allowed to move to or pass through a square that it has already visited (or passed through). The player who cannot make a move loses. Which player can guarantee a win: the one who starts or their opponent, and how should they play?

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