
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 an angle with its vertex at point $A$ and a point $B$. An arbitrary circle passing through $A$ and $B$ intersects the sides of the angle at points $C$ and $D$ (both different from $A$). Find the locus of the centroids of the triangles $ACD$.

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