
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 an arithmetic sequence $\{a\_n\}$ with a common ratio $q > 1$, and $a\_1 + a\_3 = 20$, $a\_2 = 8$.

(1) Find the general term formula of the sequence $\{a\_n\}$.

(2) Let $b\_n = \frac{n}{a\_n}$, and $S\_n$ is the sum of the first $n$ terms of the sequence $\{b\_n\}$. If the inequality $S\_n + \frac{n}{2^{n+1}} > (-1)^n \cdot a$ holds for any positive integer $n$, find the range of the real number $a$.

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