
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 ${a_n}$, define its "optimal value" $H_n$ as $H_n = \frac{a_1 + 2a_2 + \ldots + 2^{n-1}a_n}{n}$. It is known that the "optimal value" of a certain sequence is $H_n = 2^{n+1}$. Let $S_n$ denote the sum of the first $n$ terms of the sequence ${a_n - 20}$. Find the minimum value of $S_n$.

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