
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
Both triangles, right triangle $ABC$ (with hypotenuse $\overline{AB}$) and equilateral triangle $PQR$ have a shared vertex. Given that $PC = 4$, $BP = CQ = 3$, compute $AQ$ and the combined perimeter of triangles $ABC$ and $PQR$.

[asy]
unitsize(0.8 cm);

pair A, B, C, P, Q, R;

P = (0,0);
Q = (6,0);
R = 6*dir(60);
A = Q + 3*dir(120);
B = 3*dir(60);
C = (4,0);

draw(A--B--C--cycle);
draw(P--Q--R--cycle);
draw(rightanglemark(A,C,B,10));

label("$A$", A, NE);
label("$B$", B, NW);
label("$C$", C, S);
label("$P$", P, SW);
label("$Q$", Q, SE);
label("$R$", R, N);
label("$3$", (C + Q)/2, S);
label("$4$", (C + P)/2, S);
label("$3$", (B + P)/2, NW);
[/asy]

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