# 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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