
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 $\{a_n\}$ where all terms are distinct, it satisfies $a_{2n} = 2a_n - 3$, and $a_6^2 = a_1 \cdot a_{21}$. Let $S_n$ be the sum of the first $n$ terms of the sequence. Find the maximum value of the term in the sequence $\{\frac{S_n}{2^{n-1}}\}$.

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