Let \( A = \{ (a_1, a_2, \cdots, a_n : a_i = \pm 1 \, (i = 1, 2, \cdots, n) \} \subset \mathbb{R}^n \). Prove that, for any \( X \subset A \) with \( |X| > 2^{n+1}/n \), there exist three distinct points in \( X \) that are the vertices of an equilateral triangle.