Given that in a triangle \( ABC \), the angles \( \alpha, \beta, \gamma \) are opposite the sides \( a, b, \) and \( c \) respectively, with \( \alpha = 45^\circ \), \( \beta = 60^\circ \), and \( c = 10 \). Find the length of side \( a \) using the Law of Cosines and verify the result by simplification of trigonometric expressions. Let's think step by step and output the final answer within \boxed{}.
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