I want to compute the density function of the following process: I sample a value $z$ following the Laplace distribution with mean $0$ and deviation $\sigma$ (Lap($0,\sigma$)). Therefore, I have the random variable $Z\sim Lap(0,\sigma)$. Independently, I sample $y_1,y_2$ following a uniform distribution U$[0,1]$, and I compute $x_1=z\cdot y_1$, $x_2=z\cdot y_2$ so I have the random variables $X_1=Z*Y_1$ and $X_2=Z*Y_2$ where $Z\sim Lap(0,\sigma)$ and $Y_1,Y_2\sim U[0,1]$. My goal is to obtain explicitly $f_{X_1,X_2}(x_1,x_2)$ (the joint density function).
Since $f_{X_1,X_2}(x_1,x_2)=f_{X_2|X_1}(x_2|x_1)\cdot f_{X_1}(x_1)$ I first computed $f_{X_1}(x_1)$:
$$f_{X_1}(x_1)=\begin{cases} \int_{-\infty}^{x_1}\frac{1}{s}f_{Z}(s)ds & x_1\leq 0\\ \int_{x_1}^{\infty}\frac{1}{s}f_{Z}(s)ds & x_1> 0\\ \end{cases} $$ where $f_{Z}$ is the Laplace density function. However, I'm struggling in the computation of $f_{X_2|X_1}(x_2|x_1)$
So far my strategy was to separate the case in which $x_1\neq 0$ and therefore, since $x_1=z\cdot y_1$, we have: $$f_{X_2|X_1}(x_2|x_1)=\int_{-\infty}^{\infty}f_{Y_2}(\frac{x_2}{z})\cdot f_{Z|X_1}(z|x_1)dz$$ But I did not so much progress from here. Any hint or help would be appreciated.