
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 the cells of a 20 x 20 square, there are soldiers. Ivan names a number \( d \), and Petya rearranges the soldiers such that each soldier moves a distance of at least \( d \) (the distance is measured between the centers of the old and new cells). For which \( d \) is this possible?

b) The same problem for a 21 x 21 square.

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