\[\large{3\sum _{ k=0 }^{ 9 }{ { x }^{ k } } +2\sum _{ k=10 }^{ 1209 }{ { x }^{ k } } +\sum _{ k=1210 }^{ 146409 }{ { x }^{ k } } }\]

Let \(P(x)\) denote the polynomial satisfying the equation above.

Find the smallest positive integer \(n\) for which there exist polynomials \(f, g\) with integer coefficients such that

\[ x^n-1 = (x^{16} + 1)P(x) f(x) + 11g(x) . \]

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