Expanding $\prod_{m=1}^{n-1}(1+a_m+b_m)$. Quite a complicated product in the form below occured in my research. I'm having trouble getting started in evaluating it.

Let $n\in\Bbb N\setminus \{1\}$, $(a_m)_{m\in\overline{1, n-1}}\in \Bbb R^{n-1}$, and $(b_m)_{m\in\overline{1, n-1}}\in \Bbb R^{n-1}$. Consider the product $$P:=\prod_{m=1}^{n-1}(1+a_m+b_m).$$ Expand $P$.

Thoughts:
I expect binomial coefficients to show up.
If we let $c_m:= a_m+b_m$, then $P$ becomes
$$\begin{align}
\prod_{m=1}^{n-1}(1+c_m)&= (1+c_1)\prod_{m=2}^{n-1}(1+c_m) \\
&=\prod_{m=2}^{n-1}(1+c_m)+c_1\prod_{m=2}^{n-1}(1+c_m),
\end{align}$$
which suggests that induction might work if I can guess what the expansion would look like.
I think I should be able to do this myself but I've been stuck for longer than I care to mention.
Please help :)

NB: Here $0\notin \Bbb N$.

$$\begin{align}\prod_{m=1}^{n-1}(1+c_m)&=(1+c_1)(1+c_2)\cdots(1+c_{n-1})\\&=1+\sum_{\text{cyc}}c_1+\sum_{\text{cyc}}c_1c_2+\cdots+\prod_{i=1}^{n-1}c_i\end{align}$$
