
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 terms not equal to $0$, and $S_n$ is the sum of its first $n$ terms, satisfying $a_n^2=S_{2n-1}$ for $n \in \mathbb{N}^+$. If the inequality $\frac{\lambda}{a_{n+1}} \leqslant \frac{n+8\cdot (-1)^n}{n}$ 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