
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
First, calculate the number of ways to choose three points out of the 60 available. We can choose the first point in 60 ways, the second point in 59 ways, and the third point in 58 ways, resulting in $60 \cdot 59 \cdot 58 = 205320$ variations. Since the selections of vertices ABC, ACB, CBA, CAB, BAC, BCA represent the same triangle, the number of ways to choose three points disregarding the order of selection is $205320 \div 6 = 34220$. We are only concerned about the triples of points that are collinear. There are exactly 30 such triples. Therefore, there are $34220 - 30 = 34190$ ways to construct the given triangles.

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