
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
A larger equilateral triangle $UVW$ is created, where each side is divided into three equal segments at points $P$, $Q$ for side $UV$, $R$, $S$ for side $VW$, and $T$, $Z$ for side $WU$. How many noncongruent triangles can be drawn using any three of these nine points as vertices.

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