
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 \((x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4), (x_5, y_5)\) be the vertices of a regular pentagon centered at \((0,0)\). Compute the product of all positive integers \(k\) such that the equality
$$
x_{1}^{k} + x_{2}^{k} + x_{3}^{k} + x_{4}^{k} + x_{5}^{k} = y_{1}^{k} + y_{2}^{k} + y_{3}^{k} + y_{4}^{k} + y_{5}^{k}
$$
must hold for all possible choices of the pentagon.

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