
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
Let  $a$ , $b$ , $c$  be the roots of the equation  $x^{3} - 2018x +2018 = 0$ . Let  $q$  be the smallest positive integer for which there exists an integer  $p, \, 0 < p \leq q$ , such that   $$ \frac {a^{p+q} + b^{p+q} + c^{p+q}} {p+q} = \left(\frac {a^{p} + b^{p} + c^{p}} {p}\right)\left(\frac {a^{q} + b^{q} + c^{q}} {q}\right). $$   Find  $p^{2} + q^{2}$ .

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