
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
For a given positive integer \( n \geq 2 \), find the largest constant \( c \) such that for any non-negative real numbers \( a_1, a_2, \ldots, a_n \) satisfying \( a_1 + a_2 + \cdots + a_n = n \), the following inequality holds:
\[ \frac{1}{n + c a_1^2} + \frac{1}{n + c a_2^2} + \cdots + \frac{1}{n + c a_n^2} \leq \frac{n}{n + c} \]

Color some vertices (unit square corners) of an \( n \times n \) chessboard such that any sub-board of size \( k \times k \) (where \( 1 \leq k \leq n \)) has at least one red vertex on its boundary. (The \( n \times n \) chessboard has \( n+1 \) rows and \( n+1 \) columns of vertices.)
1. Color a \( 6 \times 6 \) chessboard with at most 14 red vertices satisfying the above condition.
2. Let \( l(n) \) be the minimum number of red vertices satisfying the condition. Find the value of \( \lim_{n \rightarrow \infty} \frac{l(n)}{n^2} \).

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