
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 $\{a_n\}$ satisfying $a_1 = 1$ and $a_n + a_{n-1} = \left(\frac{1}{2}\right)^n$ (for $n \geq 2$), and define \( S_n = a_1 \cdot 2 + a_2 \cdot 2^2 + \ldots + a_n \cdot 2^n \). By analogy to the method presented in the textbook for deriving the formula for the sum of the first \( n \) terms of a geometric series, find the expression for $3S_n - a_n \cdot 2^{n+1}$.

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