
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, $a\_4=10$, and $a\_1$, $a\_2$, $a\_6$ form a geometric sequence.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. Let $b\_n=2^{a\_n}+2n$, find the sum of the first $n$ terms of the sequence $\{b\_n\}$, denoted as $S\_n$.

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