Relative to a fixed origin $O$, the point $A$ has position vector $(2\mathbf{i}-\mathbf{j}+5\mathbf{k})$, the point $B$ has position vector $(5\mathbf{i}+2\mathbf{j}+\mathbf{k})$, and the point $C$ has position vector $(\mathbf{i}+a\mathbf{j}+3\mathbf{k})$, where $a$ is a constant and $a>0$. $D$ is the point such that $\overrightarrow{AB}=\overrightarrow{BD}$.