
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
To welcome the arrival of the "May Day holiday," a certain shopping mall is carrying out a promotional activity. Customers who spend over $200$ yuan in the mall can participate in a free lottery draw. The rules of the draw are as follows: There are $10$ identical balls in an opaque box, with $2$ red balls, $3$ white balls, and $5$ yellow balls. Customers draw $2$ balls from the box without replacement, and prizes are awarded based on the colors of the balls drawn. The colors of the $2$ balls drawn by the customer are divided into the following four cases: $A$: $1$ red ball and $1$ white ball, $B$: $2$ red balls, $C$: $2$ white balls, $D$: at least one yellow ball. If the four cases correspond to first prize, second prize, third prize, and no prize in order of increasing probability of occurrence.<br/>$(1)$ Find the probability that the second ball drawn by a customer in a draw is red;<br/>$(2)$ Find the probabilities of the customer winning first, second, and third prizes, respectively;<br/>$(3)$ If three customers each draw once, and whether they win a prize is independent of each other. Let the number of winners be $X$, find the probability distribution and expectation of $X$.

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