Maximize an Almost-Factorable Expression

Algebra Level 5

Real numbers \(a,b,c,d\) satisfy

\[a^2+b^2=c^2+d^2.\]

If the maximum of the expression

\[\dfrac{20bc+14bd+20ad-14ac}{a^2+b^2+c^2+d^2}\]

can be expressed as \(\sqrt{n}\) for some positive integer \(n\), what is the value of \(n?\)

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