Triangle ABC

Geometry Level 4

In triangle \(ABC\), let \(a\), \(b\) and \(c\) be the sides corresponding to \(\angle A\), \(\angle B\) and \(\angle C\), respectively. The side lengths of \(ABC\) satisfy the following two equations: \(a - 4b + 4c = 0\) and \(a + 2b - 3c = 0\). If \(\frac{\sin \angle A}{\sin \angle B + \sin \angle C} = \frac{m}{n}\), where \(m\) and \(n\) are coprime positive integers, what is the value of \(m+n\)?

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