
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 that $(a, b)$ and $(c, d)$ are two points on the line whose equation is $x + y = px + q$, and $(c, d)$ lies on the perpendicular bisector of the line segment joining $(a, b)$ and a third point $(e, f)$, find the distance between $(a, b)$ and $(c, d)$ in terms of $a, c,$ and $p$.

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