How to calculate conditional probability involving the sum of a Poisson and a Binomial random variable?

Question

I have the following process: $\{X_0, X_1, X_2, \ldots\}$

Here, $X_0 = 0$.

$X_{n+1} = P_n + B_n$

where $B_n \sim \operatorname{Binomial}(X_n, 1-p)$ and $P_n \sim \operatorname{Poisson}(1)$ independent of everything else.

This means that $(B_n \mid X_n = x_n) \sim\operatorname{Binomial} (x_n, 1-p)$

I want to verify that the process is a Markov chain. For this, I am trying to calculate:

$\mathsf P(X_k = x_k \mid X_1 = x_1, \ldots, X_{k-1} = x_{k-1})$

Here, $X_k = P_{k-1} + B_{k-1}$

So, to calculate the conditional probability, should I find the distribution of the sum of a Poisson and a Binomial? Any advice on how to proceed...? Thank you.