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[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

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[ACTION NAME]

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# Your action step 2

# Your action step 3

...

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Eating zongzi during the Dragon Boat Festival is a traditional custom in China. Suppose there are 6 zongzi in a plate, including 4 with egg yolks and 2 with red bean paste. These three types of zongzi have the same appearance. Three zongzi are randomly selected from the plate.  
$(1)$ Find the probability that at least one of the three selected zongzi is red bean paste zongzi.  
$(2)$ Let $X$ represent the number of red bean paste zongzi selected. Find the probability distribution of $X$ and the mathematical expectation $E(X)$.

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