Why Are There So Many Powers?

Algebra Level 4

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

\[x^4\left(f(x)\right)^4+x^2=3x^3\left(f(x)\right)^2,\]

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

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