
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
A certain shopping mall is conducting a lottery to attract customers. In the lottery box, there are $5$ small balls of the same size, including $2$ red balls and $3$ white balls. The rules are as follows: each time a customer randomly draws a ball from the lottery box, if a red ball is drawn, the customer wins and the ball is not put back; if a white ball is drawn, the customer loses and the ball is put back into the bag. Each customer can draw three times. Find:<br/>$(1)$ The probability that the first ball drawn is red given that the second ball drawn is white;<br/>$(2)$ The distribution and mathematical expectation of the number of red balls drawn after three draws.

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