
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 an increasing arithmetic sequence $\{a_n\}$ with the first term being $1$, and $S_n$ is the sum of its first $n$ terms, and it is known that $\frac{1}{S_1} + \frac{1}{S_2} + \frac{1}{S_3} = \frac{3}{2} (n \in \mathbb{N}^*)$.  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) Let $b_n = a_n \cdot 2^{a_n}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$.

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