
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 company carried out a salary reform, using work efficiency as an important criterion for determining salary levels, greatly improving the enthusiasm of employees. However, this also caused dissatisfaction among some senior employees. In order to investigate the relationship between salary and seniority of employees, the Human Resources Department randomly selected $16$ employees from the company's technical research and development department to understand the situation. The results are as follows:

| Seniority (years) | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ | $13$ | $14$ | $15$ | $16$ |
|-------------------|-----|-----|-----|-----|-----|-----|-----|-----|-----|------|------|------|------|------|------|------|
| Annual Salary (in 10,000s) | $9.95$ | $10.12$ | $9.96$ | $9.96$ | $10.01$ | $9.92$ | $9.98$ | $10.04$ | $10.26$ | $9.91$ | $10.13$ | $10.02$ | $9.22$ | $10.04$ | $10.05$ | $9.95$ |

After calculation, it was found that $\overline{x}=\frac{1}{16}\sum_{i=1}^{16}{x_i}=9.97$, $s=\sqrt{\frac{1}{16}\sum_{i=1}^{16}{(x_i-\overline{x})^2}}\approx0.212$, $\sqrt{\sum_{i=1}^{16}{(i-8.5)^2}}\approx18.439$, and $\sum_{i=1}^{16}{(x_i-\overline{x})(i-8.5)}=-2.78$, where $x_i$ represents the annual salary for a seniority of $i$ years, $i=1,2,\ldots,16$.

$(1)$ Calculate the correlation coefficient $r$ between the annual salary $x_i$ and seniority $i$ $(i=1,2,\ldots,16)$, and determine whether it can be considered that the annual salary and seniority have a linear relationship (if $|r|<0.25$, it can be considered that the annual salary and seniority do not have a linear relationship).

$(2)$ Among the $16$ employees selected, if the annual salary is within $(\overline{x}-3s,\overline{x}+3s)$, the salary reform will continue, and corresponding subsidies will be given to each senior employee. If an employee's annual salary is outside $(\overline{x}-3s,\overline{x}+3s)$, the employee will be interviewed by the Human Resources Department for job adjustment, and the mean and standard deviation of the annual salaries of the remaining employees among the originally selected $16$ employees need to be recalculated (rounded to $0.01$).

Given: The correlation coefficient $r$ of the sample $(x_i,y_i)$ $(i=1,2,\ldots,16)$ is $r=\frac{\sum_{i=1}^n{(x_i-\overline{x})(y_i-\overline{y})}}{\sqrt{\sum_{i=1}^n{(x_i-\overline{x})^2}}\sqrt{\sum_{i=1}^n{(y_i-\overline{y})^2}}}$, $\sqrt{0.008}\approx0.09$, $0.212^2+9.97^2\approx99.446$, $15\times10.02=1506.006$, and $9.22^2\approx85.008$.

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