Is This Just Another Boring Parabola Problem?

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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