matsh

The picture below shows a figure in the complex plane, consisting of two circles of radius 1, with centers (3,0) and (−3,0) and a lower half of the circle of radius 2 with the center (0,−23√).


Find the number of positive integers n≤1000, such that for some complex number a the equation zn−a=0 has a root on each of the three parts of the figure.

Details and assumptions
The three parts of the figure the question refers to are the two circles and the semi-circle.

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