Completeness of two statistics with one-parameter pmfs

Let $X$ and $Y$ be two random variables of discrete type with respective probability mass functions as

$$p_X(0)=2p, p_X(1)=2p, p_X(2)=1-4p,\quad0<p<\tfrac14,$$

and

$$p_Y(0)=\tfrac p2, p_Y(1)=4p^2, p_Y(2)=1-\tfrac p2-4p^2,\quad0<p<\tfrac14.$$

Then, among statistics $X$ and $Y$,