
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-filling game on a $5 \times 5$ grid where A starts and they take turns filling empty squares. Each time A fills a square with a 1, and each time B fills a square with a 0. After the grid is completely filled, the sum of the numbers in each $3 \times 3$ square is calculated, and the highest sum among these is recorded as $A$. A aims to maximize $A$, while B aims to minimize $A$. What is the maximum value of $A$ that A can achieve?

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