Length of AC makes me cry!!!???

Geometry Level 5

Let $$I$$ be the incenter of a triangle $$ABC$$ and let$$AI,BI$$ and $$CI$$ intersect $$BC,CA$$ and AB at $$D,E$$ and F respectively. If $$AB=20$$,$$BC=14$$ and $$AC=\frac{438}{53}$$, then $$\frac{ID}{IF}$$=$$\frac{a}{b}$$ for positive coprime integers $$a$$ and $$b$$. Find the value of $$a+b$$.

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