
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  $(a, b, c, d)$  of real numbers satisfying the system
                                            $(a + b)(a^2 + b^2) = (c + d)(c^2 + d^2)$                                             $(a + c)(a^2 + c^2) = (b + d)(b^2 + d^2)$                                             $(a + d)(a^2 + d^2) = (b + c)(b^2 + c^2)$ (Slovakia)

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