
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  $S$  be the set of all ordered triples  $(p,q,r)$  of prime numbers for which at least one rational number  $x$  satisfies  $px^2+qx+r=0.$  Which primes appear in seven or more elements of  $S?$ 

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