
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  $m$  and  $n$  be positive integers such that  $\gcd(m,n)=1$  and   $$ \sum_{k=0}^{2020} (-1)^k {{2020}\choose{k}}  \cos(2020\cos^{-1}(\tfrac{k}{2020}))=\frac{m}{n}. $$   Suppose  $n$  is written as the product of a collection of (not necessarily distinct) prime numbers. Compute the sum of the members of this collection. (For example, if it were true that  $n=12=2\times 2\times 3$ , then the answer would be  $2+2+3=7$ .)

*Proposed by Ankit Bisain*

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