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Algebra Level 4

\[A = \dfrac{12 x^{3} y + 108x y^{3} + 81y^{4} + x^{4} + 1 + 2y + y^{2} + 54x^{2} y^{2} }{x^{2} + x^{2} y + 6xy + 9y^{3} + 6xy^{2} + 9y^{2}}\]

Let \(T\) be the minimum value of \(A\) for positive real numbers \(x,y\). Find the value of \(5T\).

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