Discrete variables from MGFs of X and X+Y

Let $X$ and $Y$ be two independent discrete random variables such that the moment generating functions of $X$ and $X + Y$ are given by $M_X(t) = \frac{1 + 2e^{-t} + 3e^{2t}}{6}, \; t \in \mathbb{R}$, and $M_{X+Y}(t) = \frac{2 + e^{t} + 3e^{3t}}{6}, \; t \in \mathbb{R}$, respectively. Then which of the following statements is correct?