
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 $S\_n$. The sequence satisfies $a\_1=1$ and $a_{n+1} = 2S\_n + 1$ for all positive integers $n$.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. Determine the sum of the first $n$ terms, $T\_n$, for the sequence $\{ \frac{2n+1}{a\_n} \}$.

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