Find the smallest positive real number $k$ such that the following inequality holds $$\left|z_1 + \ldots + z_n\right| \geq \frac{1}{k}\left(\left|z_1\right| + \ldots + \left|z_n\right|\right)$$ for every positive integer $n \geq 2$ and every choice $z_1, \ldots, z_n$ of complex numbers with non-negative real and imaginary parts. [Hint: First find $k$ that works for $n = 2$. Then show that the same $k$ works for any $n \geq 2$.]