
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 $\{a_n\}$, which satisfies $a_1=1$ and $a_n + a_{n+1} = \left(\frac{1}{4}\right)^n$ for $n \in \mathbb{N}^*$. Let $S_n = a_1 + a_2\cdot4 + a_3\cdot4^2 + \ldots + a_n\cdot4^{n-1}$. By analogy with the method derived in the textbook for the sum of the first $n$ terms of a geometric sequence, find $5S_n - 4^n\cdot a_n$.

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