
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 $3 \times 3$ table is initially filled with zeros. In one move, any $2 \times 2$ square in the table is chosen, and all zeros in it are replaced with crosses, and all crosses with zeros. Let's call a "pattern" any arrangement of crosses and zeros in the table. How many different patterns can be obtained as a result of such moves? Patterns that can be transformed into each other by a $90^\circ$ or $180^\circ$ rotation are considered different.

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