
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 $7$ different books to be distributed among three people, A, B, and C.<br/>$(1)$ If one person gets $1$ book, one person gets $2$ books, and one person gets $4$ books, how many different ways are there to distribute the books?<br/>$(2)$ If one person gets $3$ books and the other two people each get $2$ books, how many different ways are there to distribute the books?

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