Algebra Level 5

$f(a, b, c, d) = {\frac { 1 }{ { a }^{ 2 }+1 } +\frac { 1 }{ { b }^{ 2 }+1 } +\frac { 1 }{ { c }^{ 2 }+1 } +\frac { 1 }{ { d }^{ 2 }+1 } }$

Let $$a,b,c,d$$ to be non negative real numbers satisfying $$ab+ac+ad+bc+bd+cd=6$$. Let $$\min f(a, b, c, d) = M$$.

How many ordered quadruples of non-negative real numbers $$(a, b, c, d)$$ satisfy $$f(a,b,c,d) = M$$?

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