Ridiculous length!

Level pending

In acute \(\triangle ABC\),\(AB<AC\) and \(\angle A = 60^{\circ}\) , \(H\) is the orthocentre, \(I\) is the incentre and \(D\) is the midpoint of \(BC\). The line passing through \(D\) and \(I\) intersects the straight line through \(A\) parallel to \(BC\) at \(P\). Suppose \(\sin \angle B - \sin \angle C = \dfrac{3}{5}\) and \(AH = 10\) .What is the numerical value of \(AP\) ?

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