
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
A sequence $\{a_n\}$ has a sum of its first $n$ terms denoted as $S_n$, where $a_1=1$ and $a_{n+1}=2S_n+1$ for all $n\in\mathbb{N}^*$.

(1) Find the general formula for the $n$-th term of the sequence.
(2) Let $b_n = 2n \cdot a_n$ and $T_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. Find $T_n$.

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