
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 sequence  $(a_1, a_2,\ldots, a_k)$  consisting of pairwise distinct squares of an  $n\times n$  chessboard is called a *cycle* if  $k\geq 4$  and squares  $a_i$  and  $a_{i+1}$  have a common side for all  $i=1,2,\ldots, k$ , where  $a_{k+1}=a_1$ . Subset  $X$  of this chessboard's squares is *mischievous* if each cycle on it contains at least one square in  $X$ .

Determine all real numbers  $C$  with the following property: for each integer  $n\geq 2$ , on an  $n\times n$  chessboard there exists a mischievous subset consisting of at most  $Cn^2$  squares.

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