
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
On a $5 \times 5$ board, two players alternately mark numbers on empty cells. The first player always marks 1's, the second 0's. One number is marked per turn, until the board is filled. For each of the nine $3 \times 3$ squares, the sum of the nine numbers on its cells is computed. Denote by \( A \) the maximum of these sums. How large can the first player make \( A \), regardless of the responses of the second player?

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