Is This Just Another Boring Parabola Problem?

Geometry Level 2

A parabola whose vertex is the point \(V=(2,3)\) and whose focus is \((5,6)\) has equation \(ax^2+bxy+cy^2+dx+ey+f=0\), where \(\gcd(a,b,c,d,e,f)=1\).

Find \(\big|a+b+c+d+e+f\big|.\)

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