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2023 AIME II P8 all over again

Source: 2024 AIME II Problem 13

February 8, 2024
AIMEAIME IAIME II

Problem Statement

Let ω≠1\omega \ne 1 be a 1313th root of unity. Find the remainder when ∏k=012(2−2ωk+ω2k) \prod_{k=0}^{12} \left(2 - 2\omega^k + \omega^{2k} \right) is divided by 10001000.
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