
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
Consider the graph in 3-space of
$$
0 = xyz(x+y)(y+z)(z+x)(x-y)(y-z)(z-x).
$$
This graph divides 3-space into \( N \) connected regions. What is \( N \)?

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