
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 that the sum of the first $n$ terms, $S_n$, of the sequence $\{a_n\}$ is represented by $S_n = 2a_n - n$.
1. Find the general formula for the $n^{th}$ term of the sequence $\{a_n\}$.
2. Let $b_n = \frac{2^n}{a_n a_{n+1}}$, and let $T_n$ represent the sum of the first $n$ terms of the sequence $\{b_n\}$. Find the smallest positive integer $m$ such that $T_n < \frac{m}{20}$ holds true for all $n \in \mathbb{N}^*$.

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