
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
During the Dragon Boat Festival in China, eating zongzi is a traditional custom. A plate contains 8 zongzi, including 2 red bean paste zongzi and 6 date zongzi. These two types of zongzi look identical. Three zongzi are randomly selected from the plate.  
$(1)$ Find the probability of selecting both red bean paste zongzi and date zongzi.  
$(2)$ Let $X$ represent the number of red bean paste zongzi selected. Find the probability distribution and mathematical expectation of the random variable $X$.

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