
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
In a convex quadrilateral $ABCD$, side $\overline{CD}$ is perpendicular to diagonal $\overline{AC}$, and side $\overline{AB}$ is perpendicular to diagonal $\overline{BD}$. Given that $CD = 75$, and $AB = 40$. The line through $B$ perpendicular to side $\overline{AD}$ intersects diagonal $\overline{AC}$ at $P$ with $AP = 12$. Find the length of $PC$.

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