
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
Triangle $ABC$ has a right angle at $B$, and contains a point $P$ such that $\angle BPC = 90^\circ$. Given that $PA = 12$ and $PB = 8$, find $PC$.

[asy]
unitsize(0.2 cm);

pair A, B, C, P;

A = (0,20);
B = (0,0);
C = (24,0);
P = intersectionpoint(arc(B,8,0,180),arc(C,16,0,180));

draw(A--B--C--cycle);
draw(A--P);
draw(B--P);
draw(C--P);

label("$A$", A, NW);
label("$B$", B, SW);
label("$C$", C, SE);
label("$P$", P, NE);
[/asy]

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