
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  $\omega_1, \omega_2, \omega_3, \ldots, \omega_{2020!}$  be the distinct roots of  $x^{2020!} - 1$ . Suppose that  $n$  is the largest integer such that  $2^n$  divides the value   $$ \sum_{k=1}^{2020!} \frac{2^{2019!}-1}{\omega_{k}^{2020}+2}. $$   Then  $n$  can be written as  $a! + b$ , where  $a$  and  $b$  are positive integers, and  $a$  is as large as possible. Find the remainder when  $a+b$  is divided by  $1000$ .

*Proposed by vsamc*

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