
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 $\{a\_n\}$ with distinct terms, where $a\_4 = 10$, and $a\_1$, $a\_2$, $a\_6$ form a geometric sequence. If $b\_n = 2^{a\_n} + 2n$, 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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