
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
Let $\{a_n\}$ be a sequence, and define the "excellence value" $H_0$ as $H_0= \dfrac{a_1 + 2a_2 + \ldots + 2^{n-1}a_n}{n}$. Given that the "excellence value" $H_0=2^{n+1}$ for a certain sequence, let the sum of the first $n$ terms of the sequence $\{a_n - 20\}$ be $S_n$, find the minimum value of $S_n$.

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