
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]


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A certain county's education bureau selects 3 teachers from 6 teachers recommended by county schools to participate in a training activity. Among these 6 teachers, there are 2 Chinese teachers, 2 math teachers, and 2 English teachers.  
$(1)$ Find the probability that the number of math teachers selected is greater than the number of Chinese teachers selected.  
$(2)$ Let $X$ represent the number of math teachers selected out of the 3 people. Find the distribution table and expectation of $X$.

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