
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 of an arithmetic sequence $\{a\_n\}$ is $S\_n$, and $S\_4 = 4S\_2$, $a\_2n = 2a\_n + 1$.

1. Find the general term formula of the sequence $\{a\_n\}$.
2. If the sequence $\{b\_n\}$ satisfies $\frac{b\_1}{a\_1} + \frac{b\_2}{a\_2} + \cdots + \frac{b\_n}{a\_n} = 1 - \frac{1}{2^n}$, $n \in \mathbb{N}^*$, find the sum of the first $n$ terms of $\{b\_n\}$, denoted as $T\_n$.

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