
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
We have some identical paper squares which are black on one side of the sheet and white on the other side.  We can join nine squares together to make a  $3$  by  $3$  sheet of squares by placing each of the nine squares either white side up or black side up.  Two of these  $3$  by  $3$  sheets are distinguishable if neither can be made to look like the other by rotating the sheet or by turning it over.  How many distinguishable  $3$  by  $3$  squares can we form?

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