
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
2500 chess kings have to be placed on a  $100 \times 100$  chessboard so that**(i)** no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);**(ii)** each row and each column contains exactly 25 kings.

Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.)

*Proposed by Sergei Berlov, Russia*

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