Solve the following math problem efficiently and clearly:

- For simple problems (2 steps or fewer):
Provide a concise solution with minimal explanation.

- For complex problems (3 steps or more):
Use this step-by-step format:

## Step 1: [Concise description]
[Brief explanation and calculations]

## Step 2: [Concise description]
[Brief explanation and calculations]

...

Regardless of the approach, always conclude with:

Therefore, the final answer is: $oxed{answer}$. I hope it is correct.

Where [answer] is just the final number or expression that solves the problem.

Problem: Right triangle $ABC$ (hypotenuse $\overline{AB}$) is inscribed in equilateral triangle $PQR,$ as shown.  If $PC = 3$ and $BP = CQ = 2,$ compute  $AQ.$

[asy]
unitsize(0.8 cm);

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

P = (0,0);
Q = (5,0);
R = 5*dir(60);
A = Q + 8/5*dir(120);
B = 2*dir(60);
C = (3,0);

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("$2$", (C + Q)/2, S);
label("$3$", (C + P)/2, S);
label("$2$", (B + P)/2, NW);
[/asy]
user