Find the characteristic function of Y where Y|X=x $\in N(0, x)$ with X $\in Po(\lambda)$

Question

In particular $Var(Y|X=x)=x$. I think the solution should be the characteristic function for the $N(0,\lambda)$ distribution i.e. $\varphi_y(t)=e^{-\frac{1}{2}t^2\lambda}$ but I can't figure out why the last step fails to prove this.

The related questions help slightly but not enough unfortunately as I am new to this topic.

Using probability or moment generating functions to find the distribution when given the distribution with random parameters

Compound Distribution — Normal Distribution with Log Normally Distributed Variance

My attempt:

$\varphi_y(t)=E(e^{itY})=E(E(e^{itY}|M))=E(e^{-\frac{1}{2}t^2X})=\psi_x(-\frac{1}{2}t^2)=e^{\lambda(e^{-\frac{1}{2}t^2}-1)}$

From here I don't see that $e^{\lambda(e^{-\frac{1}{2}t^2}-1)}$ belongs to any special distribution. Does this mean that this is a dead end and I have to use a different method? Thanks!