
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
Given the equation \(x^{2} + p x + q = 0\), where \(p\) and \(q\) are integers with greatest common divisor \(r(>1)\), and the roots are real numbers. Consider the sum of the \(n\)-th powers of the roots, where \(n\) is a natural number. For which powers \(k\) can we assert that the sum is divisible by \(r^{k}\)?

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