
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 rectangular chessboard of size \( m \times n \) is composed of unit squares (where \( m \) and \( n \) are positive integers not exceeding 10). A piece is placed on the unit square in the lower-left corner. Players A and B take turns moving the piece. The rules are as follows: either move the piece any number of squares upward, or any number of squares to the right, but you cannot move off the board or stay in the same position. The player who cannot make a move loses (i.e., the player who first moves the piece to the upper-right corner wins). How many pairs of integers \( (m, n) \) are there such that the first player A has a winning strategy?

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