
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
Consider an arithmetic sequence $\{a_n\}$ satisfying $a_1 = 1$ and $a_n > 0$ for $n \in \mathbb{N}^*$. Let $S_n$ denote the sum of the first $n$ terms. If the sequence $\{\sqrt{S_n}\}$ is also an arithmetic sequence, determine the maximum value of $\frac{S_{n+10}}{a_n^2}$.

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