MathNet 2011 Question 1

Let $m$ be a positive integer. Determine the smallest positive integer $n$ for which there exist real numbers $x_1, x_2, \dots, x_n \in (-1, 1)$ such that $|x_1| + |x_2| + \dots + |x_n| = m + |x_1 + x_2 + \dots + x_n|$.