
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 is known that $n$, $a\_n$, and $S\_n$ form an arithmetic sequence. Additionally, $b\_n = 2\log\_2(1 + a\_n) - 1$.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. If the terms in sequence $\{b\_n\}$ that are also in sequence $\{a\_n\}$ are removed, and the remaining terms form a new sequence $\{c\_n\}$, find the value of $c\_1 + c\_2 + ... + c\_{100}$.

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