
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 \( x_{1}, x_{2}, \cdots, x_{n} \) be a set of numbers where each value is either 1 or -1, and \( x_{1} x_{2} + x_{2} x_{3} + \cdots + x_{n-1} x_{n} + x_{n} x_{1} = 0 \). Then \( \mathrm{i}^{n} \) equals ______ .

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