
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
Ms. Carr revises her reading list and now asks her students to read any $6$ of the $12$ books provided. If Harold and Betty each randomly select $6$ books from this list, find the probability that there are exactly $3$ books that they both select.
A) $\frac{112}{617}$  
B) $\frac{1}{10}$  
C) $\frac{35}{198}$  
D) $\frac{1}{4}$  
E) $\frac{25}{154}$

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