
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
Investigate the roots of the quadratic equation $ax^2 + bx + c = 0$, where $a = p+q$, $b = p-q$, and $c = p \cdot q$ for some real numbers $p$ and $q$ and one of the roots is known to be $1$.
**A)** $\frac{-p}{p+2}$
**B)** $\frac{-2p}{p+2}$
**C)** $\frac{2p}{p-2}$
**D)** $\frac{-2p}{p-2}$
**E)** $\frac{p}{p-2}$

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