
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
Suppose that  $x^2+px+q$  has two distinct roots  $x=a$  and  $x=b$ . Furthermore, suppose that the positive difference between the roots of  $x^2+ax+b$ , the positive difference between the roots of  $x^2+bx+a$ , and twice the positive difference between the roots of  $x^2+px+q$  are all equal. Given that  $q$  can be expressed in the form  $\frac{m}{m}$ , where  $m$  and  $n$  are relatively prime positive integers, compute  $m+n$ .


*2021 CCA Math Bonanza Lightning Round #4.1*

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