
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 $z_1, z_2, \dots, z_{10}$ be the ten complex roots of the equation 
\[z^{10} - 1z^9 + 4z^8 - 9z^7 + 16z^6 - 25z^5 + 36z^4 - 49z^3 + 64z^2 - 81z + 100 = 0.\]
Calculate $\cot \left( \sum_{k = 1}^{10} \operatorname{arccot} z_k \right)$.

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