Is This Just Another Boring Parabola Problem?

Geometry Level 2

A parabola whose vertex is the point V=(2,3)V=(2,3) and whose focus is (5,6)(5,6) has equation ax2+bxy+cy2+dx+ey+f=0ax^2+bxy+cy^2+dx+ey+f=0, where gcd(a,b,c,d,e,f)=1\gcd(a,b,c,d,e,f)=1.

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

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