
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 and it satisfies $a_{2n}=2a_{n}-3$, and $a_{6}^{2}=a_{1}\cdot a_{21}$, the sum of the first $n$ terms of $\{a_n\}$ is $S_n$. Then, the maximum value among the terms of the sequence $\left\{ \frac{S_n}{2^{n-1}} \right\}$ is ______.

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