Weird Inequality, Weird Condition

Algebra Level 5

\(x,y\) are reals satisfying \[x^2+y^2=\left(\dfrac{x}{y}+\dfrac{y}{x}\right)^2\] The double-sided inequality \[m\le \dfrac{x^3y^3+x^2y+xy^2+1}{x^3y^3}\le M\] is always true, where \(m\) is the largest possible and \(M\) is the smallest possible. Find \[\lfloor 1000\{M+m\}\rfloor \] where \(\{x\}\) means the fractional part of \(x\) (i.e \(\{x\}=x-\lfloor x\rfloor\))

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