AMC 10 2022b Problem 10

Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode? $\textbf{(A)}\ 5 \textbf{(B)}\ 7 \textbf{(C)}\ 9 \textbf{(D)}\ 11 \textbf{(E)}\ 13$