
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
The "Wisdom" Bookstore of the school recruits around 30 volunteers for the freshmen of Grade 11 at the beginning of each school year. At the beginning of the 2021 school year, the new Grade 11 students signed up enthusiastically, with a total of 60 people registering. There are two plans to determine the volunteers: Plan 1: randomly select 30 volunteers by drawing lots; Plan 2: number the 60 registrants, randomly select 45 numbers from these 60 numbers using a random number method, and then randomly select another 45 numbers from these 60 numbers. Those who are selected in both rounds become volunteers.

$(1)$ Using Plan 1 or Plan 2, let the event that student A is selected be event $A$ and the event that student B is selected be event $B$. Find the probabilities of events $A$ and $B$ occurring.

$(2)$ It is not difficult to find that the number of volunteers determined by Plan 2 is no less than 30 from Plan 1. If Plan 2 is adopted, let the number of people selected in both rounds be $Y$. What are the possible values of $Y$? Which value of $Y$ is the most likely to occur?

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