
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

Given real numbers \(a_{1}, a_{2}, \cdots, a_{n}\) which are all greater than 0 (where \(n\) is a natural number no less than 4) and satisfy the equation \(a_{1} + a_{2} + \cdots + a_{n} = 1\), find the maximum value of the sum \(S = \sum_{k=1}^{n} \frac{a_{k}^{2}}{a_{k} + a_{k+1} + a_{k+2}} \quad\text{with}\ (a_{n+1} = a_{1}, a_{n+2} = a_{2})\).

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