Weird Inequality, Weird Condition

Algebra Level 5

x,yx,y are reals satisfying x2+y2=(xy+yx)2x^2+y^2=\left(\dfrac{x}{y}+\dfrac{y}{x}\right)^2 The double-sided inequality mx3y3+x2y+xy2+1x3y3Mm\le \dfrac{x^3y^3+x^2y+xy^2+1}{x^3y^3}\le M is always true, where mm is the largest possible and MM is the smallest possible. Find 1000{M+m}\lfloor 1000\{M+m\}\rfloor where {x}\{x\} means the fractional part of xx (i.e {x}=xx\{x\}=x-\lfloor x\rfloor)

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