
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
$A$ and $B$ are playing a game on a $25 \times 25$ grid. Initially, $A$ can mark some squares. After marking, they take turns placing coins on the grid, with $B$ placing the first coin. The rules for placing the coins are as follows: (1) Coins cannot be placed in marked squares; (2) Once a coin is placed, no coins can be placed in the same row or column as that coin; (3) The game ends when no more coins can be placed on the grid. The player who places the last coin wins. How many squares does $A$ need to mark at the beginning to ensure they win the game, assuming both $A$ and $B$ play optimally?

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