\[\large \begin{cases} (a+b)(c+d) = 143 \\ (a+c)(b+d) = 150 \\ (a+d)(b+c) = 169 \end{cases} \]

Let \(a\), \(b\), \(c\), and \(d\) be positive real numbers which satisfy the system of equations above. Find the smallest possible value of \(a^2 + b^2 + c^2 + d^2\).

Note: Full credit is given to the makers of this problem.

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