
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 during its annual year-end summary meeting. In an opaque box, there are $3$ red balls and $3$ white balls (the balls are of the same shape and size). The rules of the lottery are as follows: draw $3$ balls from the bag at once, turn the white balls into red balls, and put them all back into the bag. Let $X$ be the number of red balls in the bag at this time. Each employee will be awarded a bonus of $X$ thousand yuan.

$(1)$ Find the distribution table and mathematical expectation of $X$;
$(2)$ 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 $100$ thousand yuan and the variance is $225$ thousand yuan, to motivate employees who make outstanding contributions to the company, it is decided that only employees whose contribution exceeds $115$ thousand yuan can receive the bonus. The total bonus for rewards is calculated based on the mathematical expectation of the bonus obtained from the lottery scheme. Find the number of employees who will receive the bonus and the average amount of bonus each person will receive (rounded to the nearest integer).

Reference data: If a random variable $\xi$ follows a normal distribution $N(\mu, \sigma^2)$, then $P(\mu - \sigma < \xi \leq \mu + \sigma) \approx 0.6827$, $P(\mu - 2\sigma < \xi \leq \mu + 2\sigma) \approx 0.9545$.

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