
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
**polyhedral**

we call a  $12$ -gon in plane good whenever:
first, it should be regular, second, it's inner plane must be filled!!, third, it's center must be the origin of the coordinates, forth, it's vertices must have points  $(0,1)$ , $(1,0)$ , $(-1,0)$  and  $(0,-1)$ .
find the faces of the <u>massivest</u> polyhedral that it's image on every three plane  $xy$ , $yz$  and  $zx$  is a good  $12$ -gon.
(it's obvios that centers of these three  $12$ -gons are the origin of coordinates for three dimensions.)

time allowed for this question is 1 hour.

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