
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
There's a convex  $3n$ -polygon on the plane with a robot on each of it's vertices. Each robot fires a laser beam toward another robot. On each of your move,you select a robot to rotate counter clockwise until it's laser point a new robot. Three robots  $A$ ,  $B$  and  $C$  form a triangle if  $A$ 's laser points at  $B$ ,  $B$ 's laser points at  $C$ , and  $C$ 's laser points at  $A$ . Find the minimum number of moves that can guarantee  $n$  triangles on the plane.

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