
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
In a $101 \times 101$ grid, each cell is filled with a number from the set $\{1, 2, \cdots, 101^2\}$, and each number in the set is used exactly once. The left and right boundaries of the grid are considered the same line, and the top and bottom boundaries are also considered the same line (i.e., the grid forms a torus). Determine the maximum value of $M$ such that no matter how the grid is filled, there exist two adjacent cells (sharing a common edge) whose filled numbers have a difference of at least $M$.

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