
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
There are $8$ balls of the same size, including $4$ different black balls, $2$ different red balls, and $2$ different yellow balls.<br/>$(1)$ Arrange these $8$ balls in a line, with the black balls together, the 2 red balls adjacent, and the 2 yellow balls not adjacent. Find the number of ways to arrange them;<br/>$(2)$ Take out $4$ balls from these $8$ balls, ensuring that balls of each color are taken. Find the number of ways to do so;<br/>$(3)$ Divide these $8$ balls into three groups, each group having at least $2$ balls. Find the number of ways to divide them.

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