
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 cockroaches run along a circle in the same direction. They start simultaneously from a point  $S$ . Cockroach  $A$  runs twice as slow than  $B$ , and thee times as slow than  $C$ . Points  $X, Y$  on segment  $SC$  are such that  $SX = XY =YC$ . The lines  $AX$  and  $BY$  meet at point  $Z$ . Find the locus of centroids of triangles  $ZAB$ .

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