Let $X_t= e^{\left(\mu-\sigma^2/2 \right)t+\sigma W_t}$ be a geometric Brownian motion with drift $\mu$ and volatility $\sigma$. I am trying to derive an analytical solution to
$$\mathbb{E}\left[ \max(a X_T + b X_S -K,0)\right],$$ where $a$, $b$ and $K$ are constants and $0<S<T$.
My objective is to find the critical point as from which $aX_T + bX_S$ will be greater than $K$ so that I can disregard the maximum function and evaluate the expectation.
Am I right to say that if I had only $Y= X_T + X_S$, I could use the relation $X_T + X_S=2X_S + X_T - X_S$ to find its mean and variance and subsequently find the critical point?
Is there any way to proceed in the same way for my original problem?