
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 numbers \( z_0 = 1 - m \mathrm{i}(m > 0), z = x + y \mathrm{i} \), and \( w = x' + y' \mathrm{i} \), where \( x, y, x', y' \in \mathbf{R} \), and \( \mathrm{i} \) is the imaginary unit. For any complex number \( z \), it holds that \( w = \bar{z}_0 \cdot \bar{z} \) and \(|w| = 2|z| \).

(1) Find the value of \( m \), and express \( x' \) and \( y' \) in terms of \( x \) and \( y \).

(2) Let \( (x, y) \) be the coordinates of point \( P \), and \( (x', y') \) be the coordinates of point \( Q \). The above relationships can be considered as a transformation of points in the coordinate plane: transforming point \( P \) to point \( Q \). When point \( P \) moves along the line \( y = x + 1 \), find the equation of the locus of point \( Q \) after the transformation.

(3) Does there exist a line such that any point on it, after the above transformation, still lies on the same line? If so, find all such lines; if not, provide a reason.

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