
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
In response to the national medical and health system reform and the "Three Visits to the Countryside" cultural and scientific activities nationwide in 2023, to truly implement the concept of "putting the people first" and promote the transfer and decentralization of high-quality medical resources, continuously enhance the "depth" and "warmth" of medical services. The People's Hospital of our city plans to select 3 doctors from the 6 doctors recommended by each department to participate in the activity of "Healthy Countryside Visit, Free Clinic Warming Hearts." Among these 6 doctors, there are 2 surgeons, 2 internists, and 2 ophthalmologists.
- $(1)$ Find the probability that the number of selected surgeons is greater than the number of selected internists.
- $(2)$ Let $X$ represent the number of surgeons selected out of the 3 people. Find the mean and variance of $X$.

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