
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 $100$ books, which are divided into two categories: liberal arts and science, and two types: hardcover and softcover. Among them, there are $40$ liberal arts books, $70$ hardcover books, and $20$ softcover science books. Calculate:

$(1)$ The probability of randomly selecting a liberal arts hardcover book is $\_\_\_\_\_\_\_\_$;

$(2)$ The probability of first randomly selecting a liberal arts book, then returning it and randomly selecting a hardcover book is $\_\_\_\_\_\_\_\_$.

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