
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 $S$ be the [set](https://artofproblemsolving.com/wiki/index.php/Set) of all [points](https://artofproblemsolving.com/wiki/index.php/Point) with [coordinates](https://artofproblemsolving.com/wiki/index.php/Coordinate) $(x,y,z)$, where $x$, $y$, and $z$ are each chosen from the set $\{0,1,2\}$. How many [equilateral](https://artofproblemsolving.com/wiki/index.php/Equilateral) [triangles](https://artofproblemsolving.com/wiki/index.php/Triangle) all have their [vertices](https://artofproblemsolving.com/wiki/index.php/Vertices) in $S$?
$(\mathrm {A}) \ 72\qquad (\mathrm {B}) \ 76 \qquad (\mathrm {C})\ 80 \qquad (\mathrm {D}) \ 84 \qquad (\mathrm {E})\ 88$

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