
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 a convex $3n$-gon on a plane with a robot at each vertex, each robot shoots a laser pointing at another robot. Define an operation as selecting a robot and rotating it clockwise until its laser points at a new robot. For three robots $A$, $B$, and $C$, if robot $A$'s laser points at robot $B$, robot $B$'s laser points at robot $C$, and robot $C$'s laser points at robot $A$, these three robots form a "triangle." What is the minimum number of operations needed to ensure that $n$ triangles appear on the plane?

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