How to infer about the distribution of $Y$ from the given

Question

Let $X\sim\mathsf{Ber}\left(p\right)$. Let $Y$ be random variable such that $\left(Y|X=k\right)\sim\mathsf{Poi}(k+1)$ for $k\in\{0,1\}$. find the distribution of $Y$

my attempt:

$$P(Y|X=k)=\frac{P(Y=t,X=k)}{P(X=k)} $$ $$P(Y|X=k)=\frac{P(Y=t,X=k)}{p(1-p)}$$

using the given about $P(Y|X=k)=$ I think I can write $$P(Y|X=k)=\frac{(k+1)^n}{n!}e^{-(k+1)}$$

from here I dont know how to procced. I am not sure if I wrote the expressions for $P(Y|X=k)$ and $P(X=k)$ correctly

Answer 1

$$\mathbb{P}(Y=l) = \mathbb{P}(Y=l,X=0)+\mathbb{P}(Y=l,X=1) = \mathbb{P}(Y=l|X=0)\mathbb{P}(X=0)+\mathbb{P}(Y=l|X=1)\mathbb{P}(X=1) = (1-p)\cdot \frac{1^l}{l!}e^{-1} + p\frac{2^l}{l!}e^{-2} = \frac{e^{-1}}{l!}(p2^le^{-1}+(1-p))$$