Consider a single qubit CPTP map $\mathcal{N}$ such that $$\mathcal{N}(I) = I + pZ,~~~~~~\mathcal{N}(Z) = (1-p)Z,$$

where $I$ and $Z$ are Pauli operators. For an $n$ qubit Pauli operator $P$, made only of $I$ and $Z$, I am trying to find $\mathcal{N}^{\otimes n}(P).$

Let the weight of $P$ be $|P|$. I know that $\mathcal{N}^{\otimes n}(P)$ will be multiplied by a prefactor of $(1-p)^{|P|}$. But how do I write the rest of the terms, without cluttering the notation too much?