
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 rectangle divided by two lines parallel to its width (bisecting it into equal horizontal thirds) and three lines parallel to its length (dividing it into four equal vertical sections), how many different triangles can be found? All intersecting points between lines and edges are connected, splitting the large rectangle into numerous smaller triangles.

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