
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
Consider the following system of  $10$  equations in  $10$  real variables  $v_1, \ldots, v_{10}$ :
\[v_i = 1 + \frac{6v_i^2}{v_1^2 + v_2^2 + \cdots + v_{10}^2} \qquad (i = 1, \ldots, 10).\]
Find all  $10$ -tuples  $(v_1, v_2, \ldots , v_{10})$  that are solutions of this system.

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