If `|z_1-1|lt=,|z_2-2|lt=2,|z_(33)|lt=3,` then find the greatest value of `|z_1+z_2+z_3|dot`

Explanation

To solve the problem, we need to find the greatest value of \( |z_1 + z_2 + z_3| \) given the constraints on the complex numbers \( z_1, z_2, \) and \( z_3 \).

### Step-by-Step Solution:

1. **Understanding the Constraints**:
- We have \( |z_1 - 1| \leq 1 \), which means \( z_1 \) lies within or on the boundary of a circle centered at \( 1 \) with a radius of \( 1 \).
- We have \( |z_2 - 2| \leq 2 \), which means \( z_2 \) lies within or on the boundary of a circle centered at \( 2 \) with a radius of \( 2 \).
- We have \( |z_3| \leq 3 \), which means \( z_3 \) lies within or on the boundary of a circle centered at \( 0 \) with a radius of \( 3 \).
...