Let $ ABCDE$ be a convex pentagon such that\[ \angle BAC= \angle CAD =\angle DAE\qquad \text{and}\qquad \angle ABC =\angle ACD = \angle ADE.\]The diagonals $BD$ and $CE$ meet at $P$, $AP$ and $CD$ meet at $X$. If $DX=7$, find the value of $CD$.