
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 $\{a_n\}$ whose sum of the first $n$ terms $S_n = 2^n$, and a sequence $\{b_n\}$ satisfying $b_1 = -1$, $b_{n+1} = b_n + (2n-1)$ for $n=1,2,3,\ldots$.

(1) Find the general term $a_n$ of the sequence $\{a_n\}$;  
(2) Find the general term $b_n$ of the sequence $\{b_n\}$;  
(3) If $c_n = \frac{a_n \cdot b_n}{n}$, find the sum of the first $n$ terms $T_n$ of the sequence $\{c_n\}$.

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