\[\begin{cases} a+b+c-d = 1 \\ a+b-c+d = 2 \\ a-b+c+d = 3 \\ -a+b+c+d = 4 \end{cases} \]

If the system of equations above holds true for the positive real number of \( a, b, c, d\), then find the value of the expression below.

\[\begin{eqnarray} 8a^2 + 9b^2 + 12c^2 + 13d^2 \end{eqnarray} \]

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