Geometry or calculus? II

Calculus Level 5

Given a ellipse with equation $$2x^2+3y^2+x-y-5=0$$ and a point $$P(3,-1)$$, there are two lines passing through $$P$$ that are also tangent to the ellipse. One line is represented by the equation $$ax+by-c=0$$ and the other one is $$dx-ey-f=0$$, where $$a,b,c,d,e,f>0$$, $$\gcd(a,b,c)=1$$ and $$\gcd(d,e,f)=1$$. Find $$\dfrac{d+e+f-2}{a+b+c}$$.

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