
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  $A$  be the set of all integers  $n$  such that  $1 \le n \le 2021$  and  $\text{gcd}(n,2021)=1$ . For every nonnegative integer  $j$ , let
\[
S(j)=\sum_{n \in A}n^j.
\]
Determine all values of  $j$  such that  $S(j)$  is a multiple of  $2021$ .

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