
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
10. The sum  $\sum_{k=1}^{2020} k \cos \left(\frac{4 k \pi}{4041}\right)$  can be written in the form  $$ \frac{a \cos \left(\frac{p \pi}{q}\right)-b}{c \sin ^{2}\left(\frac{p \pi}{q}\right)} $$  where  $a, b, c$  are relatively prime positive integers and  $p, q$  are relatively prime positive integers where  $p<q$ . Determine  $a+b+c+p+q$ .

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