How to solve this complex equation in MATHEMATICA?

Question

I am trying to solve a complex integral equation including an infinite continued fraction.

Continued fraction to order n=50

$Version

(*14.2.0 for Microsoft Windows (64-bit) (December 26, 2024)*)

Clear["Global`*"]

frac[x_?NumericQ, y_?NumericQ, b_?NumericQ] := ContinuedFractionK[1,1 - 1/(x + I y) + ContinuedFractionK[-(b^2/(x + I y)^2) (t^2 - 1)/t^2 n(n + 2)/(4 (n + 1)^2- 1), 1 - 1/(x + I y), {n, 1, 50}], {n,1,1}]// ComplexExpand

Equation to be solved which contains the integral of the continued fraction for z=20

z = 20;

equation[x_?NumericQ, y_?NumericQ, b_?NumericQ] := 1 - b^2/(x + I y)^2 - z^2 /(3 BesselK[2, z] (x + I y)^2) NIntegrate[(t^2 - 1)^(3/2)/t frac[x, y, b] Exp[-z t],{t,1, Infinity}] // ComplexExpand

In order to proceed step by step, I have three questions to ask Mathematica

- First: reformulate the continued fraction frac[] as a real part Re[frac] plus an imaginary part Im[frac]

- Second: find the correct starting points for x and y for equation[] to include in FindRoot

- Third: find and plot all the real and imaginary roots x(b) and y(b) in function of the parameter b, with 0.01 =< b =< 20

roots[b_] := {x, y} /.FindRoot[{Re[equation[x, y, b]] == 0, Im[equation[x, y, b]] == 0}, {{x, 1.2}, {y, - 0.01}}]

As a beginner, this equation seems difficult to solve with mathematica, and the calculation time for a given b is very long.

How can I control the code more efficiently in order to obtain the best roots for this problem?