
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 play a number-changing game on a $5 \times 5$ grid: A starts and both take turns filling empty spaces, with A filling each space with the number 1 and B filling each space with the number 0. After the grid is completely filled, the sum of the numbers in each $3 \times 3$ square is calculated, and the maximum sum among these squares is denoted as $A$. A tries to maximize $A$, while B tries to minimize $A$. What is the maximum value of $A$ that A can achieve? 

(The problem is from the 35th IMO selection test)

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