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Number Theory Level 4

The function \(f:\mathbb{N}\to\mathbb{N}\) satisfies: \[m^2+f(n)\mid mf(m)+n ~~~\forall ~(m,n)\in\mathbb{N}^2.\] Let the sum of all possible values of \(f(2014)\) be \(N\). Find the value of \(N\bmod{1000}\).

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