
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
Select $5$ people from $7$ people including $A$, $B$, and $C$ to line up in a row (all answers should be numerical).<br/>$(1)$ If $A$ must be included, how many ways are there to line them up?<br/>$(2)$ If $A$, $B$, and $C$ are not all included, how many ways are there to line them up?<br/>$(3)$ If $A$, $B$, and $C$ are all included, $A$ and $B$ must be adjacent, and $C$ is not adjacent to $A$ or $B$, how many ways are there to line them up?

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