
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
Three arbitrary parallel lines \( d_{a}, d_{b}, d_{c} \) are drawn through the vertices of triangle \(ABC\). The lines symmetric to \( d_{a}, d_{b}, d_{c} \) with respect to \(BC, CA, AB\) respectively form a triangle \(XYZ\). Find the locus of the centers of the incircles of such triangles.

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