Why Are There So Many Powers?

Algebra Level 4

For a function f(x)f(x) such that

x4(f(x))4+x2=3x3(f(x))2,x^4\left(f(x)\right)^4+x^2=3x^3\left(f(x)\right)^2,

let the largest possible value of f(2014)f(2014) over all possible functions f(x)f(x) be M.M. Find 1000M.\lfloor 1000M\rfloor .

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