
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
Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$, with $a_1=1$ and $a_{n+1} = 2S_n + 1$ for all $n \in \mathbb{N}^*$.

1. Find the general term formula for the sequence $\{a_n\}$.
2. If $\frac{b_n}{a_n} = 3n - 1$, find the sum of the first $n$ terms, $T_n$, of the sequence $\{b_n\}$.

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