
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
Consider an arithmetic sequence $\{a_n\}$ where the sum of the first $n$ terms is denoted by $S_n$. It is given that $S_n = \frac{1}{2}na_n + a_n - c$ where $c$ is a constant and $n \in \mathbb{N}^*$. If $a_2 = 6$, and another sequence $\{b_n\}$ is defined by $b_n = \frac{a_n - 2}{2^{n+1}}$, with the sum of the first $n$ terms of $\{b_n\}$ denoted by $T_n$. If $2T_n > m - 2$ holds for all $n \in \mathbb{N}^*$, find the maximum value of the positive integer $m$.

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