
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 a circle \((x-14)^2 + (y-12)^2 = 36^2\) and a point \(C(4,2)\) inside the circle. Two points \(A\) and \(B\) that lie on the circumference of the circle are such that \(\angle ACB = 90^\circ\). Find the equation of the locus of the midpoint of the hypotenuse \(AB\).

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