
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
In the three-dimensional rectangular coordinate system, \( E \) is the set of triangles with vertices at integral points and without any other integral points either inside or on the edges. Find the set of areas \( f(E) \) of these triangles.
(Context: The 30th International Mathematical Olympiad Candidate Problems, 1989)

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