
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 square piece of paper with 2005 points inside, along with the four vertices of the square forming a set \( M \), where no three points are collinear, the paper is to be completely cut into triangles according to the following rules:
1. Each triangle's vertices must be points from \( M \).
2. Aside from the vertices, each triangle must not contain any other points from \( M \).

How many triangles can be cut out in total? How many cuts are needed? (Each edge of a triangle requires one cut.)

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