Given $\mathbb E(5X+2)=12$ and $\mathbb E(X|Y)=Y^3$ compute $\mathbb E(Y^3)$

Question

Given $\mathbb E(5X+2)=12$ and $\mathbb E(X|Y)=Y^3$ compute $\mathbb E(Y^3)$.

I've been trying this a million different ways and can't seem to reach a final answer, I would love any suggestions!

Answer 1

$$\mathbb E(5X+2)=5\,\mathbb E(X)+2=12 \implies \mathbb E(X)=2$$

$$\mathbb E(Y^3)=\mathbb E\bigl(\mathbb E(X|Y)\bigr)=\mathbb E(X)=2$$

Answer 2

$E[X] = 2$.

Since $E[ X|Y] = Y^3$, then $E[Y^3] = E[ E[X|Y]] = E[X] = 2$.