
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 $(x_1, y_1)$ be a point on the line $y = mx + k$ and $(x_2, y_2)$ be a point on the parabola $y = ax^2 + bx + c$. Given that $x_1 = p$ and $x_2 = q$, derive the distance between these two points in terms of $p, q, m, a, b,$ and $c$.

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