
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
A six dimensional "cube" (a  $6$ -cube) has  $64$  vertices at the points  $(\pm 3,\pm 3,\pm 3,\pm 3,\pm 3,\pm 3).$   This  $6$ -cube has  $192\text{ 1-D}$  edges and  $240\text{ 2-D}$  edges.  This  $6$ -cube gets cut into  $6^6=46656$  smaller congruent "unit"  $6$ -cubes that are kept together in the tightly packaged form of the original  $6$ -cube so that the  $46656$  smaller  $6$ -cubes share 2-D square faces with neighbors ( $\textit{one}$  2-D square face shared by  $\textit{several}$  unit  $6$ -cube neighbors).  How many 2-D squares are faces of one or more of the unit  $6$ -cubes?

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