In a triangle \(ABC\), a circle centred at \(O\), passing through \(B\) and \(C\), is drawn such that \( \angle BOC = 144^{\circ}\) where \(O\) is on the side of \(BC\) not containing \(A\). The circle intersects \(AB\) and \(AC\) again at \(P\) and \(Q\) respectively. It is given that the incentre \(I\) of \(APQ\) lies on the circle.

Find \(\angle BAO\).

This problem is part of the set Advent Calendar 2014.

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