
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 \( z_1, z_2, \dots, z_{21} \) be the twenty-one (complex) roots of the equation
\[
z^{21} - 5z^{20} + 16z^{19} - 49z^{18} + \dots + 576 = 0.
\]
Calculate \(\cot \left( \sum_{k = 1}^{21} \operatorname{arccot} z_k \right)\).

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