
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
For each prime \( p \), a polynomial \( P(x) \) with rational coefficients is called \( p \)-good if and only if there exist three integers \( a, b \), and \( c \) such that \( 0 \leq a < b < c < \frac{p}{3} \) and \( p \) divides all the numerators of \( P(a) \), \( P(b) \), and \( P(c) \), when written in simplest form. Compute the number of ordered pairs \((r, s)\) of rational numbers such that the polynomial \( x^{3}+10x^{2}+rx+s \) is \( p \)-good for infinitely many primes \( p \).

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