
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 color the numbers  $1, 2, 3,....,20$  with two colors white and black in such a way that both colors are used. Find the number of ways, we can perform this coloring if the product of white numbers and the product of black numbers have greatest common divisor equal to  $1$ .

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