
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
Suppose \( x_{1}, x_{2}, \cdots, x_{n}, y_{1}, y_{2}, \cdots, y_{n}, z_{1}, z_{2}, \cdots, z_{n} \) are all 1 or -1, and \( x_{1} y_{1}+x_{2} y_{2}+\cdots+x_{n} y_{n}=0 \), \( x_{1} z_{1}+x_{2} z_{2}+\cdots+x_{n} z_{n}=0 \), \( y_{1} z_{1}+y_{2} z_{2}+\cdots+y_{n} z_{n}=0 \). Find the value of \( n \).

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