Let $ABCD$ be a convex quadrilateral with $AB = AD$, $m\angle A = 40^{\circ}$, $m\angle C = 130^{\circ}$, and $m\angle ADC - m\angle ABC = 20^{\circ}$. Find the measure of the non-reflex angle $\angle CDB$ in degrees.

Let's think step by step and output the final answer within \boxed{}.
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