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We are given the equation involving a non-zero complex number \( z \) which satisfies:

\[
\frac{\text{Re}\left(z + \bar{z} + |z|\right)}{\text{Im}\left(z - \bar{z} - i |z|\right)} = 1,
\]
where \( i = \sqrt{-1} \) is the imaginary unit and \( \bar{z} \) denotes the complex conjugate of \( z \).

### Step-by-step Analysis:

Let \( z = x + iy \) where \( x \) and \( y \) are the real and imaginary parts of \( z \), respectively. Therefore, the following holds:
- \( \bar{z} = x - iy \) (complex conjugate of \( z \)),
- \( |z| = \sqrt{x^2 + y^2} \) (modulus of \( z \)).

Substitute these into the equation:

#### 1. Real part:
\[
\text{Re}(z + \bar{z} + |z|) = \text{Re}(x + iy + x - iy + |z|) = \text{Re}(2x + |z|).
\]
Thus, the real part is:
\[
2x + |z| = 2x + \sqrt{x^2 + y^2}.
\]

#### 2. Imaginary part:
\[
\text{Im}(z - \bar{z} - i|z|) = \text{Im}(x + iy - (x - iy) - i|z|) = \text{Im}(2iy - i|z|) = 2y - |z|.
\]

#### 3. Equation:
Substitute these expressions into the original equation:

\[
\frac{2x + \sqrt{x^2 + y^2}}{2y - \sqrt{x^2 + y^2}} = 1.
\]

#### 4. Solving the equation:
Cross-multiply to get:
\[
2x + \sqrt{x^2 + y^2} = 2y - \sqrt{x^2 + y^2}.
\]

Move terms involving \( \sqrt{x^2 + y^2} \) to one side:
\[
2x + \sqrt{x^2 + y^2} + \sqrt{x^2 + y^2} = 2y,
\]
\[
2x + 2\sqrt{x^2 + y^2} = 2y,
\]
\[
x + \sqrt{x^2 + y^2} = y.
\]

Square both sides to eliminate the square root:
\[
\left(x + \sqrt{x^2 + y^2}\right)^2 = y^2,
\]
\[
x^2 + 2x\sqrt{x^2 + y^2} + x^2 + y^2 = y^2.
\]
Simplify:
\[
2x^2 + 2x\sqrt{x^2 + y^2} = 0.
\]
Thus,
\[
x(x + \sqrt{x^2 + y^2}) = 0.
\]

This gives us two cases:
1. \( x = 0 \),
2. \( x + \sqrt{x^2 + y^2} = 0 \).

#### Case 1: \( x = 0 \)
If \( x = 0 \), the equation reduces to:
\[
0 + \sqrt{0^2 + y^2} = y \quad \Rightarrow \quad y = y,
\]
so no restriction on \( y \). Hence, \( z = iy \) is purely imaginary.

#### Case 2: \( x + \sqrt{x^2 + y^2} = 0 \)
This implies:
\[
\sqrt{x^2 + y^2} = -x.
\]
However, since \( \sqrt{x^2 + y^2} \geq 0 \), this case does not give valid solutions.

### Conclusion:
The set of exhaustive values of \( \arg(z) \) (argument of \( z \)) is \( \frac{\pi}{2} \) or \( \frac{3\pi}{2} \), meaning that \( z \) lies on the imaginary axis. Therefore, \( z \) is purely imaginary.

Do you have any questions or need further details?

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**Related questions:**
1. What is the geometrical interpretation of the argument of a complex number?
2. How does the modulus of a complex number influence its properties?
3. How can complex conjugates help in solving equations involving complex numbers?
4. What role does the imaginary unit \( i \) play in complex number calculations?
5. How does the polar form of a complex number simplify the calculation of arguments?

**Tip:** When solving complex number equations, expressing the complex number in terms of real and imaginary parts can significantly simplify the solution process.

Consider a non-zero complex number z satisfying the equation: Re(z + z̅ + |z|) / Im(z - z̅ - i|z|) = 1, where i = √-1. The set of exhaustive values of arg(z) is?

Explore how to solve for the argument of a complex number z satisfying Re(z + z̅ + |z|) / Im(z - z̅ - i|z|) = 1. This problem covers complex number theory, including the modulus and argument relationship, and the role of the complex conjugate.

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