
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
Each of the two distinct roots of the quadratic polynomial \( f(x) = x^2 + px + q \) and its value at \( x = 3 \) are natural numbers, where the larger root of the polynomial and \( f(3) \) are prime numbers. Find the roots of the polynomial \( f(x) \).

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