Q50

Let $\Sigma=\{1,2,3,4\}$. For $x \in \Sigma^{*}$, let $\operatorname{prod}(x)$ be the product of symbols in $x$ modulo 7. We take $\operatorname{prod}(\epsilon)=1$, where $\epsilon$ is the null string. For example, $\operatorname{prod}(124)=(1 \times 2 \times 4) \bmod 7=1$. Define $L=\left\{x \in \Sigma^{*} \mid \operatorname{prod}(x)=2\right\}$. The number of states in a minimum state DFA for $L$ is __________ . (Answer in integer)