
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 $z_1, z_2, \dots, z_{12}$ be the twelve complex roots of the equation
\[z^{12} - 3z^{11} + 6z^{10} - 10z^9 + \dots + 144 = 0.\]
Calculate $\cot \left( \sum_{k = 1}^{12} \operatorname{arccot} z_k \right).$

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