A lot of tangency points

Geometry Level 4

The triangle \(ABC\) is inscribed in the circle \(\mathcal{C}\). Circle \(\mathcal{C}_1\) is tangent to the minor arc \(AB\) of \(\mathcal{C}\) and to side \(AC\) at point \(P\). Circle \(\mathcal{C}_2\) is tangent to the minor arc \(CB\) of \(\mathcal{C}\) and to side \(AC\) at point \(Q\). From point \(B\) draw the tangents \(BR\) and \(BS\) to circles \(\mathcal{C}_1\) and \(\mathcal{C}_2\), respectively. It is known that \(BA=22\), \(BC=23\), \(BR=16\), \(BS=17\), \(PQ=15\). \(AC\) can be written as \( \frac{a}{b} \), where \(a\) and \(b\) are coprime positive integers. What is the value of \(a+b\)?

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