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

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.

$\begin{aligned} 8a^2 + 9b^2 + 12c^2 + 13d^2 \end{aligned}$