
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given the complex number \( z_{0} = 1 - m \mathrm{i} \) (where \( m > 0 \)), and \( z = x + y \mathrm{i} \) and \( w = x^{\prime} + y^{\prime} \mathrm{i} \), where \( x, y, x^{\prime}, y^{\prime} \in \mathbb{R} \), and \( \mathrm{i} \) is the imaginary unit, it is known that for any complex number \( z \), \( w = \bar{z_{0}} \cdot \bar{z} \) and \( |w| = 2|z| \).

1. Determine the value of \( m \) and express \( x^{\prime} \) and \( y^{\prime} \) in terms of \( x \) and \( y \).
2. Let \( (x, y) \) be the coordinates of point \( P \), and \( (x^{\prime}, y^{\prime}) \) be the coordinates of point \( Q \). The above relationship can be seen as a transformation of points in the coordinate plane: it transforms point \( P \) in the plane to point \( Q \) in the same plane. When point \( P \) moves along the line \( y = x + 1 \), find the equation of the trajectory of point \( Q \) after the transformation.
3. Does there exist a line such that any point on it, after undergoing the above transformation, still lies on the same line? If it exists, find all such lines; if not, provide a reason.

Present the answer in LaTex format: \boxed{Your answer}
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