
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 the sum of its first $n$ terms denoted as $S\_n$. It is known that $S\_n = \frac{1}{2}na\_n + a\_n - c$, where $c$ is a constant and $n \in \mathbb{N}^*$. Also, $a\_2 = 6$.

(I) Find the value of $c$ and the general term formula for the sequence $\{a\_n\}$.
(II) Let $b\_n = \frac{a\_n - 2}{2^{n+1}}$, and denote the sum of the first $n$ terms of the sequence $\{b\_n\}$ as $T\_n$. Find the smallest positive integer $n$ such that $T\_n > \frac{199}{100}$ always holds.

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