
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 positive integer \( k \), find the largest positive integer \( m \) such that if no more than \( m \) of any set of \( 3k \) points in the plane are collinear, then the points can be divided into \( k \) sets of three points each, where the points in each set form the vertices of a triangle.

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