Proving a Markov chain $M_n$ is the unique one such that $M_n$ and $M_nS_n$ are both Markov chains

Question

I've been stuck on this problem from my homework.

"Let $S_n$ be the position of a biased random walk on the integers $0$ to $N$ where $S_0=a$. Define $M_n := \frac{1}{[4p(1-p)]^{n/2}} \left(\frac{1-p}{p} \right)^{S_n/2}.$ Let $F_n$ denote the information of $S_0, ... ,S_n$.

1. Prove that $M_n$ is a martingale with respect to $F_n$.
2. Prove that $M_nS_n$ is a martingale with respect to $F_n$.
3. Suppose that $R_n$ is a process such that $R_0 = M_0$ and both $R_n$ and $R_nS_n$ are martingales with respect to $F_n$. Show $R_n = M_n$ for all $n$."

I can do 1 and 2 fine, but I'm completely stuck on 3. Can anyone help me?