
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 $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$, it satisfies the equation $2S\_n = a_{n+1} - 2^{n+1} + 1$, where $n \in \mathbb{N}^*$. Additionally, $a\_1$, $a\_2 + 5$, and $a\_3$ form an arithmetic sequence. Determine the expression for $a\_n$.

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