
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 placed numbers from 1 to $99^2$ in each cell of a $99 \times 99$ board, using each number exactly once. Petya chooses a cell, places a chess king on it, and wants to make as many moves as possible with the king such that the number beneath it constantly increases. What is the maximum number of moves that Petya can certainly make, regardless of how Vasya arranges the numbers?

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