
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 $\{a_n\}$ with the general term $a_n = (2n-1) \times 2^n$, find the sum of the first $n$ terms, $S_n$. For this, we use the method of subtraction by displacement, which can be shown as follows: From
$$
S_n = 1 \times 2 + 3 \times 2^2 + 5 \times 2^3 + \ldots + (2n-1) \times 2^n
$$
we get
$$
2S_n = 1 \times 2^2 + 3 \times 2^3 + 5 \times 2^4 + \ldots + (2n-1) \times 2^{n+1}
$$
Subtracting the two equations, we obtain
$$
-S_n = 2 + 2 \times 2^2 + 2 \times 2^3 + \ldots + 2 \times 2^n - (2n-1) \times 2^{n+1}
$$
Solving for $S_n$, we find
$$
S_n = 2 - (2-2n) \times 2^{n+1}
$$
Using a similar method, for a sequence $\{b_n\}$ with the general term $b_n = n^2 \times 2^n$, find the sum of the first $n$ terms $T_n$.

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