Applying Monte Carlo to a multi-level problem

Question

Here is my problem statement, and the bolded item in the bulleted list is where I'm having trouble with. I think my question comes from my poor understanding of Monte Carlo sampling and conditional probability.

Problem Statement: I need to perform Monte Carlo simulation for a problem where, in its simplest form, can be represented as:

$Z= f(Y1(\mu,\sigma), Y2(\mu,\sigma))$, where

$\mu_{Y1} = f(X1(\mu,\sigma), X2(\mu,\sigma))$

I know the distributions of $X1$, $X2$, and $Y2$, and also $\sigma_{Y2}$, and I want to apply Monte Carlo to get realizations of $Z$.

Flow of analysis:

1. Generate $N$ samples on $X1$, $X2$, and $Y2$
2. Calculate an instance of $\mu_{Y1}$ for every pair of $X1$ and $X2$ samples
3. What to do here? For every $\mu_{Y1}$ instance, do I
   • generate 1 sample of $Y1$ and propagate this sample with the sample of $Y2$ to get $Z$, or
   • generate $N$ samples of $Y1$, which increases the dimension of $Z$ to $N^2$?
4. Obtain my realizations of $Z$

Am I even on the right track? Any help and thoughts are much appreciated.