From a circular piece of paper with radius \( BC \), Jeff removes the unshaded sector shown. Using the larger shaded sector, he joins edge \( BC \) to edge \( BA \) (without overlap) to form a cone of radius 12 centimeters and of volume \( 432\pi \) cubic centimeters. What is the number of degrees in the measure of angle \( ABC \) of the sector that is not used? Let's think step by step and output the final answer within \boxed{}.
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