Suppose $a$, $b$, $c$, and $d$ are positive real numbers that satisfy the system of equations:
\[
\begin{align*}
(a+b)(c+d) &= 143, \\
(a+c)(b+d) &= 150, \\
(a+d)(b+c) &= 169.
\end{align*}
\]
Compute the smallest possible value of $a^2+b^2+c^2+d^2$.

Let's think step by step and output the final answer within \boxed{}.
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