# 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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