
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
After studying the content of "High School Mathematics Compulsory $3$", a student in the second year of high school believes that there is a correlation between exam scores and the number of exams. So he collected the scores of his first $5$ exams since entering the second year of high school, as listed in the table below:

| Exam Number | 1 | 2 | 3 | 4 | 5 |
|-------------|---|---|---|---|---|
| Exam Score  | 80| 95| 95|100|105|

After further research, he found that the exam score $y$ is linearly related to the number of exams $x$.

$(1)$ Find the linear regression equation of $y$ with respect to $x$, $\hat{y}=\hat{b}x+\hat{a}$;

$(2)$ Determine whether the variables $y$ and $x$ are positively or negatively correlated (just write the conclusion);

$(3)$ According to the plan, there are a total of $10$ exams in the two semesters of the second year of high school. Please predict the student's score for the last exam in the second year of high school (round to the nearest whole number).

Given: The least squares estimation formulas for the slope and intercept of the regression line are $\hat{b}=\frac{{\sum_{i=1}^n{{x_i}{y_i}-n\overline{x}\overline{y}}}}{{\sum_{i=1}^n{x_i^2-n{{\overline{x}}^2}}}}=\frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})({{y_i}-\overline{y}})}}}{{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}}}}$ and $\hat{a}=\overline{y}-\hat{b}\overline{x}$.

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