
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
Given that \( z_{1}, z_{2}, \ldots, z_{7} \) are the seventh roots of \( 2021 + i \), and their principal arguments in ascending order are \( \alpha_{1}, \alpha_{2}, \ldots, \alpha_{7} \), find the value of \( \tan \alpha_{1} \tan \alpha_{3}+\tan \alpha_{2} \tan \alpha_{4}+\cdots+\tan \alpha_{7} \tan \alpha_{2} \).

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