
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) In all cells of a $20 \times 20$ square, there are soldiers. Ivan names a number $d$, and Peter rearranges the soldiers so that each one moves a distance of at least $d$ (the distance is measured between the centers of the old and new cells). For which values of $d$ is this possible?

b) Solve the same problem for a $21 \times 21$ square.

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