What is the distribution of sum of a Gaussian and and 2 r.v. Rayleigh distributed?

Question

Let $Z=X+Y+W$;

where $X∼N(0,σ_1^2)$ i.e. a Gaussian random variable and Y and W follow the Rayleigh distribution:

$f_w(w)=\frac{w}{σ_2^2} . exp(−\frac{w^2}{2σ_2^2})$, $y\ge0$

What will be the distribution of Z, assuming X,Y and W indipendent?

I read another post very similar in this forum: What is the distribution of sum of a Gaussian and a Rayleigh distributed independent r.v.?

and I would only have the confirmation that I could achieve the $Z$'s pdf applying two times the convolution rule. I mean first convolution between $f_y$ and $f_w$, then $f_{yw}$ against $f_x$.

Thanks in advance.