Let a, b, c, and d be positive real numbers which satisfy the system of equations
\begin{aligned}
& (a+b)(c+d)=143, \\
& (a+c)(b+d)=150, \\
& (a+d)(b+c)=169.
\end{aligned}
Find the smallest possible value of a^{2}+b^{2}+c^{2}+d^{2}.
Let a, b, c, and d be positive real numbers which satisfy the system of equations
Find the smallest possible value of a^{2}+b^{2}+c^{2}+d^{2}.