Question:

Each constraint defines a disk: $|z_1+1| \leq 1$ means $z_1$ lies in a disk centered at $-1$ with radius $1$, so $|z_1| \leq |z_1+1| + 1 \leq 1 + 1 = 2$. Similarly, $|z_2+2| \leq 2$ gives $|z_2| \leq |z_2+2| + 2 \leq 2 + 2 = 4$, and $|z_3+4| \leq 4$ gives $|z_3| \leq |z_3+4| + 4 \leq 4 + 4 = 8$. Using the reverse triangle inequality optimally: when $z_k$ is positioned on the ray opposite to the center of each disk, we get $|z_1|_{max} = 2$, $|z_2|_{max} = 4$, and $|z_3|_{max} = 8$. Therefore, $|z_1| + |z_2| + |z_3|_{max} = 2 + 4 + 8 = 14$.