
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 $w_1,$ $w_2,$ $\dots,$ $w_{10}$ be the ten (complex) roots of the equation
\[w^{10} - 2w^9 + 4w^8 - 8w^7 + \dots + 256 = 0.\]Calculate $\cot \left( \sum_{k = 1}^{10} \operatorname{arccot} w_k \right).$

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