A curve passing through the point \((1,1)\) has the property that the perpendicular distance of the origin from the normal at any point \(P\) of the curve is equal to the distance of \(P\) from the \(x\)-axis .

Let the equation of the curve be of the form

\[ax^2 +by^2 + 2hxy +2gx+2fy+c = 0\]

for integers \(a,b,c,g,h\) and \(f\). where \(a,b\) are coprime positive integers

Calculate \(a+b+2g\).

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