Find the complex number z for which `|z+1| = z + 2(1 + i)`.

Explanation

To find the complex number \( z \) for which \( |z + 1| = z + 2(1 + i) \), we will follow these steps:

### Step 1: Express \( z \) in terms of its real and imaginary parts
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 2: Rewrite the equation
The given equation is:
\[
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