
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 all possible quadratic trinomials \(x^{2} + px + q\) with a positive discriminant, where the coefficients \(p\) and \(q\) are integers divisible by 5. Find the largest natural number \(n\) such that for any trinomial with these properties, the sum of the hundredth powers of the roots is an integer divisible by \(5^{n}\).

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