
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
\( ABC \) is a triangle with coordinates \( A = (2, 6), B = (0, 0) \), and \( C = (14, 0) \).
(a) Let \( P \) be the midpoint of \( AB \). Determine the equation of the line perpendicular to \( AB \) passing through \( P \).
(b) Let \( Q \) be the point on line \( BC \) for which \( PQ \) is perpendicular to \( AB \). Determine the length of \( AQ \).
(c) There is a (unique) circle passing through the points \( A, B \), and \( C \). Determine the radius of this circle.

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