Melanie computes the mean $\mu$, the median $M$, and the modes of the $365$ values that are the dates in the months of $2019$. Thus her data consist of $12$ $1$s, $12$, $2$s, $\cdots$, $12$ $28$s, $11$ $29$s, $11$ $30$s, and $7$ $31$s. Let $d$ be the median of the modes. Which of the following statements is true?

$(A)$ $\mu < d < M$ $\qquad$ $(B)$ $M < d < \mu$ $\qquad$ $(C)$ $d=M=\mu$ $\qquad$ $(D)$ $d < M <\mu$ $\qquad$ $(E)$ $d < \mu < M$