
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
In a bag, there are $5$ balls of the same size, including $3$ red balls and $2$ white balls.<br/>$(1)$ If one ball is drawn with replacement each time, and this process is repeated $3$ times, with the number of times a red ball is drawn denoted as $X$, find the probability distribution and expectation of the random variable $X$;<br/>$(2)$ If one ball is drawn without replacement each time, and the color is recorded before putting it back into the bag, the process continues until two red balls are drawn, with the number of draws denoted as $Y$, find the probability of $Y=4$.

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