
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
On the sides of triangle \(ABC\), points were marked: 10 on side \(AB\), 11 on side \(BC\), and 12 on side \(AC\). None of the vertices of the triangle were marked. How many triangles with vertices at the marked points exist?

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