Let \(P_1, P_2, \ldots , P_{2009} \) be the points on the side \(BC\) of a square \(ABCD\) so that \(BP_1 = P_1 P_2 = P_2 P_3 = \cdots = P_{2008} P_{2009} = P_{2009} C \).

Let \(Q\) be a point on the side \(AD\) so that \(AQ = BP_1 \).

Find the sum of angles \(\angle AP_1 Q + \angle AP_2 Q + \cdots + \angle AP_{2009} Q + \angle ACQ \) in degrees.

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