
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
In an $n \times n$ grid of small squares, if a sequence of $k$ distinct small squares $\left(a_{1}, a_{2}, \ldots, a_{k}\right)$ satisfies: $k \geq 4$, and each small square $a_{i}$ shares an edge with $a_{i+1}$ $\left(i=1,2, \ldots, k\right.$, and $\left.a_{k+1}=a_{1}\right)$, then the sequence $\left(a_{1}, a_{2}, \ldots, a_{k}\right)$ is called a cycle. If a set $X$ of some small squares in the grid satisfies that for every cycle in the grid, at least one small square is in $X$, then set $X$ is called a "naughty set."

Find all real numbers $C$ such that for any integer $n \geq 2$, there exists a "naughty set" in an $n \times n$ grid containing at most $C n^{2}$ small squares.

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