
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 company held a lottery at an annual year-end meeting. In an opaque box, there are $3$ red balls and $3$ white balls (the shape and size of the balls are the same). There are two options for the lottery:<br/>Option 1: Select an employee to randomly draw a ball from the bag. If it is a red ball, put it back in the bag; if it is a white ball, do not put it back and then add a red ball to the bag. Repeat this process $3$ times. If the final number of red balls in the bag is $X$, each employee will receive a bonus of $X$ thousand yuan;<br/>Option 2: Draw $3$ balls from the bag at once, replace the white balls with red balls, and put all the balls back into the bag. If the number of red balls in the bag is $Y$, each employee will receive a bonus of $Y$ thousand yuan.<br/>$(1)$ If option 1 is used, find the distribution and mathematical expectation of $X$;<br/>$(2)$ Compare option 1 with option 2, and determine which option has a higher mathematical expectation for the amount of bonus employees can receive. Explain the reason;<br/>$(3)$ If the company has 1000 employees and their contributions to the company's profits approximately follow a normal distribution $N(\mu, \sigma^2)$, where the mean contribution of each employee is calculated to be $100$ thousand yuan and the variance is calculated to be $225$ thousand yuan. If it is stipulated that only employees who contribute profits greater than $115$ thousand yuan can receive a bonus, calculate the maximum value of the mathematical expectation of the bonus obtained by employees using option 1 and option 2. Determine the number of winning employees and the average amount of bonus each person can receive (rounded to the nearest integer).<br/>Reference data: If a random variable $\xi$ follows a normal distribution $N(\mu, \sigma^2)$, then $P(\mu - \sigma < \xi \leq \mu + \sigma) \approx 0.6826$.

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