Show Geometric Brownian motion is the unique solution to $\frac{dS_t}{S_t}=\mu dt+\sigma dW_t$.

Question

Given $$ X_t=\sigma W_t+\mu t, $$ the SDE $$ \frac{dS_t}{S_t}=\mu dt+\sigma dW_t $$ has a unique solution $$ Z_t=\exp\Big(X_t-\frac{1}{2}\langle X\rangle_t\Big). $$ We know from various posts and textbooks (e.g. Deriving Geometric Brownian Motion's solution?) that $Z_t$ is a solution. To show uniqueness, I want to apply Ito's formula to $R_t=Z_t'/Z_t$, where $Z_t$,$Z_t'$ are two solutions, then $$ dR_t=-\frac{Z_t'}{Z_t^2}dZ_t+\frac{1}{Z_t}dZ_t'+\frac{Z_t'}{Z_t^3}d\langle Z\rangle_t-\frac{1}{Z_t^2}d\langle Z,Z' \rangle_t. $$ The first two terms cancel out as $dZ_t=Z_tdX_t$, but how can we deal with the last two terms? In particular, is it possible to conclude some relationships between $d\langle Z,Z' \rangle_t$ and $d\langle X \rangle_t$?