
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 a nonconstant polynomial with real coefficients  $f(x),$  let  $S(f)$  denote the sum of its roots. Let p and q be nonconstant polynomials with real coefficients such that  $S(p) = 7,$   $S(q) = 9,$  and  $S(p-q)= 11$ . Find, with proof, all possible values for  $S(p + q)$ .

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