
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 quadratic function $f(x) = ax^2 + bx$ whose graph passes through the point $(-4n, 0)$, and $f'(0) = 2n$ ($n \in \mathbb{N}^*$),
1. Find the explicit formula for $f(x)$.
2. Let the sequence $\{a_n\}$ be defined as $a_n = f'(-n) \cdot 2^n$, find the sum of the first $n$ terms of the sequence $\{a_n\}$.

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