
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
Vasya has placed numbers from 1 to \(101^2\) in each cell of a \(101 \times 101\) board, using each number exactly once. Petya selects a cell on the board and places a piece there. Petya wants to make as many moves as possible, such that the number under the piece always increases. In one move, Petya can move the piece to any cell within a \(5 \times 5\) square centered on the current cell (the piece must stay within the board). What is the maximum number of moves Petya can definitively make, regardless of how Vasya placed the numbers?

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