
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
A point T is placed on line segment AB such that the sum of the distances from A to T and B to T combined is increased by 3 units from previously $p$ units and the product of these distances is now $s^2$ (from $q^2$). Find the quadratic polynomial whose roots are the distances AT and BT.
A) $x^2 - px + s^2$
B) $x^2 - (p+3)x + s^2$
C) $x^2 - (p+3)x + q^2$
D) $x^2 - px + q^2$

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