Strange Inequality

Algebra Level 5

A function \(f:\mathbb{R}^2 \rightarrow \mathbb{R}\) is defined as follows: \[f(x, y)= \sqrt{x^2+y^2-12x-14y+85} \] Another function \(g:\mathbb{R}^2 \rightarrow \mathbb{R}\) is defined as follows: \[g(x, y)= \sqrt{x^2+y^2-10x-16y+89} \] As \(a, b, c, d\) range over all reals (not necessarily positive) such that \[\begin{cases} f(a,b)g(a,b)f(c,d)g(c,d) & \neq 0 \\ a+b & < 13 \\ c+d & > 13, \end{cases}\] let \(M\) be the minimum value of \[\dfrac{f(a,b)g(c,d)+f(c,d)g(a,b)}{\sqrt{a^2+b^2+c^2+d^2-2ab-2cd}}.\] Find \(M^2.\)

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