
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
Express the polynomial \( f(x) = 1 - x + x^2 - x^3 + \cdots - x^{19} + x^{20} \) as a polynomial in \( y \), \( g(y) = a_0 + a_1 y + a_2 y^2 + \cdots + a_{19} y^{19} + a_{20} y^{20} \), where \( y = x - 4 \). Then calculate \( a_0 + a_1 + \cdots + a_{20} \).

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