A wonky quartic

Algebra Level 5

The sum of all values of \(a\) such that the equation \[(x^2-x+a+1)^2=4a(5x^2-x+1)\] has exactly three distinct real solutions, is of the form \(\dfrac{n+\sqrt{k}}{m}\), where \(k,m,n\) are integers, \(k\geq 0,\) \(m\geq 1\) and \(m\) is the smallest possible. Find \(k+n+m.\)

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