
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 sequence $\{a_n\}$ where $a_1=1$, $a_2=3$, and for any $n\in N^*$, $a_{n+2}\leqslant a_{n}+3\cdot2^{n}$, $a_{n+1}\geqslant 2a_{n}+1$ always holds, then the sum of the first $n$ terms of the sequence $\{a_n\}$, $S_n=$ ______.

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