
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 that the sequence $\{a_n\}$ is a non-zero arithmetic sequence, and $S_n$ is the sum of its first $n$ terms, with the condition that $a_n = \sqrt{S_{2n-1}}$ for $n \in \mathbb{N}^*$. If the inequality $\frac{\lambda}{a_n} \leqslant \frac{n+8}{n}$ holds for any $n \in \mathbb{N}^*$, then the maximum value of the real number $\lambda$ is ______.

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