
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 an arithmetic sequence with all non-zero terms, and \\(S_n\) is the sum of its first \\(n\) terms, and it is known 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}
user