
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
Consider a  $25\times25$  chessboard with cells  $C(i,j)$  for  $1\le i,j\le25$ . Find the smallest possible number  $n$  of colors with which these cells can be colored subject to the following condition: For  $1\le i<j\le25$  and for  $1\le s<t\le25$ , the three cells  $C(i,s)$ ,  $C(j,s)$ ,  $C(j,t)$  carry at least two different colors.

(Proposed by Gerhard Woeginger, Austria)

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