
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 19th Asian Games were successfully held in Hangzhou from September 23 to October 8, 2023. The volunteers of the Hangzhou Asian Games are called "Little Green Lotus." There are a total of 36 Little Green Lotus in a sports venue, including 12 boys and 24 girls. The number of Little Green Lotus who can speak Japanese and Korean is as follows:

|              | Boys Little Green Lotus | Girls Little Green Lotus |
|--------------|-------------------------|--------------------------|
| Can speak Japanese | 8 | 12 |
| Can speak Korean | m | n |

where m and n are positive integers, and 6 ≤ m ≤ 8.

$(1)$ Two Little Green Lotus are randomly selected from these 36 individuals to serve as hosts for an event. Find the probability that at least one of the two selected Little Green Lotus can speak Japanese.

$(2)$ One Little Green Lotus is randomly selected from these individuals to greet foreign guests. Let A represent the event "the selected Little Green Lotus is a boy" and B represent the event "the selected Little Green Lotus can speak Korean." Provide a set of values for m and n such that events A and B are independent, and explain the reasoning.

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