
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 quadruplets  $(x_1, x_2, x_3, x_4)$  of real numbers such that the next six equalities apply:  $$ \begin{cases}  x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4
x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4
x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3
x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4
x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 
x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2  \end{cases} $$  

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