
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 $\triangle ABC$, $AB = 10$, $BC = 6$, $CA = 8$, and side $AB$ is extended to a point $P$ such that $\triangle PCB$ is similar to $\triangle CAB$. Find the length of $PC$.
[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair A=origin, P=(1.5,5), B=(10,0), C=P+2.5*dir(P--B);
draw(A--P--C--A--B--C);
label("A", A, W);
label("B", B, E);
label("C", C, NE);
label("P", P, NW);
label("8", 3*dir(A--C), SE);
label("6", B+3*dir(B--C), NE);
label("10", (5,0), S);
[/asy]

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