What Is This Line?

Geometry Level 4

In \(\triangle ABC,\) \(AB= 6, BC= 7, CA= 5.\) There exists a unique line \(\ell\) passing through \(A\) such that reflections of \(B,C\) about \(\ell\) lie on lines \(CA, AB\) respectively. Suppose \(\ell\) intersects \(BC\) at point \(D.\) If \(\dfrac{BD}{DC}= \dfrac{a}{b}\) for some coprime positive integers \(a, b,\) find \(a+b.\)

Details and assumptions
- The reflections of \(B,C\) in \(\ell\) lie on the lines \(CA, AB\) respectively. They might not lie on the segments \(CA, AB.\)
- In the diagram above, \(B', C'\) are the reflections of \(B,C\) respectively.
- The diagram shown is not accurate.

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