Juggling Points On An Incircle
Let \(ABC\) be a triangle and let \(\omega\) be its incircle.
Denote by \(D_1\) and \(E_1\) the points where \(\omega\) is tangent to sides \(BC\) and \(AC\), respectively. Denote by \(D_2\) and \(E_2\) the points on sides \(BC\) and \(AC\), respectively, such that \(CD_2=BD_1\) and \(CE_2=AE_1\), and denote by \(P\) the point of intersection of segments \(AD_2\) and \(BE_2\).
Circle \(\omega\) intersects segment \(AD_2\) at two points, the closer of which to the vertex \(A\) is denoted by \(Q\). Given that \(D_2P = 5\), find \(AQ.\)