
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
With the completion of the Rongcheng Ecological Park Greenway, cycling around the city has become one of the most popular outdoor leisure activities. The total length of the city's greenway is about 100 kilometers, which not only allows cyclists to circle around Rongcheng but also provides a 360-degree panoramic view of the city's development and charm. A student has cycled 5 times in the past half year, and the comprehensive physical index score $x$ and corresponding time $y$ (in hours) for each cycling session are shown in the table below:

| Physical Index Score ($x$) | 1   | 2   | 3   | 4   | 5   |
|---------------------------|-----|-----|-----|-----|-----|
| Time ($y$ in hours)       | 9.5 | 8.6 | 7.8 | 7.0 | 6.1 |

$(1)$ Based on the data in the table, a linear regression model can be used to fit the relationship between $y$ and $x$. Please explain using the correlation coefficient.  
$(2)$ Establish the regression equation of $y$ with respect to $x$.  
Reference data and formulas:  
Correlation coefficient $r=\frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})}({{y_i}-\overline{y}})}}{{\sqrt{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}}\sum_{i=1}^n{{{({{y_i}-\overline{y}})}^2}}}}$, $\hat{b}=\frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})}({{y_i}-\overline{y}})}}{{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}}}$, $\hat{a}=\overline{y}-\hat{b}\overline{x}$, $\sqrt{7060}≈84$.

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