
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 a geometric sequence $\{a_n\}$ with a common ratio $q > 1$, and $a_1 + a_3 = 20$, $a_2 = 8$.

(Ⅰ) Find the general formula for the sequence $\{a_n\}$;

(Ⅱ) Let $b_n = \frac{n}{a_n}$, and $S_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. For any positive integer $n$, the inequality $S_n + \frac{n}{2^{n+1}} > (-1)^n \cdot a$ always holds. Find the range of the real number $a$.

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