AMC 10 2021b Problem 13

Let $n$ be a positive integer and $d$ be a digit such that the value of the numeral $\underline{32d}$ in base $n$ equals $263$, and the value of the numeral $\underline{324}$ in base $n$ equals the value of the numeral $\underline{11d1}$ in base six. What is $n + d ?$ $\textbf{(A)} ~10 \textbf{(B)} ~11 \textbf{(C)} ~13 \textbf{(D)} ~15 \textbf{(E)} ~16$