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: An equilateral triangle $PQR$ is inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$ so that $Q$ is at $(0,b),$ and $\overline{PR}$ is parallel to the $x$-axis, as shown below.  Also, foci $F_1$ and $F_2$ lie on sides $\overline{QR}$ and $\overline{PQ},$ respectively.  Find $\frac{PQ}{F_1 F_2}.$

[asy]
unitsize(0.4 cm);

pair A, B, C;
pair[] F;
real a, b, c, s;

a = 5;
b = sqrt(3)/2*5;
c = 5/2;
s = 8;

A = (-s/2,-sqrt(3)/2*(s - 5));
B = (0,b);
C = (s/2,-sqrt(3)/2*(s - 5));
F[1] = (c,0);
F[2] = (-c,0);

draw(yscale(b)*xscale(a)*Circle((0,0),1));
draw(A--B--C--cycle);

label("$P$", A, SW);
label("$Q$", B, N);
label("$R$", C, SE);
dot("$F_1$", F[1], NE);
dot("$F_2$", F[2], NW);
[/asy]
user