
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 the sequence $\{a_n\}$ with the general term formula $a_n = (2n-1) \cdot 2^n$, we want to use the method of staggered subtraction to find the sum of the first $n$ terms, $S_n$. From $S_n = 1 \cdot 2 + 3 \cdot 2^2 + 5 \cdot 2^3 + \ldots + (2n-1) \cdot 2^n$, we get $2S_n = 1 \cdot 2^2 + 3 \cdot 2^3 + 5 \cdot 2^4 + \ldots + (2n-1) \cdot 2^{n+1}$. By subtracting term-by-term, we arrive at: $-S_n = 2 + 2 \cdot 2^2 + 2 \cdot 2^3 + \ldots + 2 \cdot 2^n - (2n-1) \cdot 2^{n+1}$, and we find that $S_n = (2n-3) \cdot 2^{n+1} + 6$. By analogy, if the sequence $\{b_n\}$ has the general term formula $b_n = n^2 \cdot 2^n$, what is the sum of the first $n$ terms, $T_n$?

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