
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, $S\_n$ denotes the sum of its first $n$ terms, and satisfies $a\_n^2 = S\_{2n-1}$ for all positive integers $n$. If the inequality $\frac{λ}{a\_{n+1}} \leqslant \frac{n + 8 \cdot (-1)^n}{2n}$ holds true for any positive integer $n$, determine the maximum value of the real number $λ$.

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