AMC 12 2015a Problem 19

For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible?

\textbf{(A)}\ 30 \textbf{(B)}\ 31 \textbf{(C)}\ 61 \textbf{(D)}\ 62 \textbf{(E)}\ 63