
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
Consider the sum   $$ S =\sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|. $$  The value of  $S$  can be written as  $\tan \left( \frac{c\pi}{d} \right)$  for some relatively prime positive integers  $c, d$ , satisfying  $2c < d$ . Find the value of  $c + d$ .

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