
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
Find all real solutions \( x_1, x_2, \ldots, x_n \) to the equation

\[ x_i^3 (x_i^2 + x_{i+1}^2 + \ldots + x_{i+k-1}^2) = x_{i-1}^2 \]

for \( i = 1, 2, \ldots, n \), where \( k \) and \( n \) are positive integers such that \( n > k > 1 \), and \( x_0 \) means \( x_n \), and \( x_{n+j} \) means \( x_j \).

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