
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 the arithmetic sequence $\{a_{n}\}$ with $a_{1}=1$ and $a_{n} > 0$ for all $n \in \mathbb{N}^*$, let $S_{n}$ denote the sum of its first $n$ terms. If the sequence $\{\sqrt{S_{n}}\}$ is also an arithmetic sequence, find the maximum value of $\frac{S_{n+10}}{a_{n}^{2}}$.

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