
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
Let \(T\) be the set of all points \( (x,y,z) \) where each coordinate \( x, y, z \) is selected from the set \( \{0,1,2,3\} \) and the sum \( x+y+z \) is even. Find the number of equilateral triangles with all their vertices in \( T \).

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