Consider a parallelogram constructed on the vectors $\overrightarrow{A}=5\overrightarrow{p}+2\overrightarrow{q}$ and $\overrightarrow{B}=\overrightarrow{p}-3\overrightarrow{q}.$ If $\left|\overrightarrow{p}\right|=2, \left|\overrightarrow{q}\right|=5,$ the angle between $\overrightarrow{p}$ and $\overrightarrow{q}$ is $\frac{\pi }{3}$ and the length of the smallest diagonal of the parallelogram is $k$ units, then the value of $k^2$ is equal to