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Geometry Level 5

\[ a^2 \cot 9^\circ + b^2 \cot 27^\circ + c^2 \cot 63^\circ + d^2 \cot 81^\circ \]

Let \(a,b,c\) and \(d\) be real numbers satisfying \(a+b+c+d=5\). If the minimum value of the expression above is equal to \( \dfrac{\sqrt x}y \), where \(x\) and \(y\) are coprime positive integers, find \(x+y\).

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