*ABC* is inscribed in a circle \(\Gamma_1\) so that *BC* is a diameter. Circle \(\Gamma_2\) is inscribed in *ABC*, and circle \(\Gamma_3\) is tangent to both segment *AB* and minor arc *AB* so that the two points of tangency lie on the same diameter of \(\Gamma_1\). If \(\Gamma_2\) and \(\Gamma_3\) are congruent, then the extended ratio *BC* : *AC* : *AB* can be written *a* : *b* : *c*, where *a*, *b*, and *c* are positive coprime integers. Find \(a+b+c\).

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