Let $x_1, x_2, \ldots, x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p + x_q + x_r$ (with $1 \le p < q < r \le 5$) are equal to $0$, then $x_1 = x_2 = \cdots = x_5 = 0$.

Let's think step by step and output the final answer within \boxed{}.
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