probability - Expectation of variable number of subsequent die throws

A die is thrown, and a number $x$ shows up. Then $x$ die are thrown, and the sum on all die is said to be $y$. Now $y$ die are thrown, and sum of numbers on die on each throw is said to be $z$. What is the expected value of $z$

At first this problem looks like it needs programming to solve, as individually there will be many cases. Eg, for $x = 2$, $y$ can be anything from $2$ to $12$ and then $z$ can be anything from $2$ to $72$.

A recursive program may be used to compute the probabilities for all possible $z$, which in all cases can lie from $1$ to $6^3$. The problem with brute-force computation clearly is that for larger values of $y$, number of cases on $z$ increases very quickly. For $y=36$, there will be $36$ die each with $6$ choices so a total of $6^{36}$ operations for this case itself!

This recursive approach is not suitable, but I unable to find another method to tackle this question. Another method could be counting cases manually, but that too is not very plausible.

Any clues are welcome!

Answer

$\mathbb E[Z]=\mathbb E[\mathbb EZ\mid Y]]=\mathbb E[3.5Y]]=3.5\mathbb E[Y]=3.5\mathbb E[\mathbb E[Y\mid X]]=3.5\mathbb E[3.5X]=3.5^2\mathbb EX=3.5^3$