UMVUE in a three-cell multinomial

Let $(Y_1,Y_2,Y_3)\in\{0,1,…,n\}^3$ be a discrete random vector having joint probability mass function $$\mathbb{P}(Y_1=y_1,Y_2=y_2,Y_3=y_3)=\begin{cases}\dfrac{n!}{y_1!y_2!y_3!}\,p^{y_1}(2p)^{y_2}(1-3p)^{y_3}& \text{if }y_1+y_2+y_3=n,\ 0& \text{otherwise,}\end{cases}$$ where $0\le p\le\frac13$ is an unknown parameter. Assume the convention $0^0=1$. The maximum likelihood estimator of $p$ is denoted by $\hat p$. Which of the following statements is correct?