
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
To implement rural revitalization and promote agricultural technology, a village has built a comprehensive rural complex and invited experts from the provincial capital to provide technical guidance. According to statistics, the corresponding data between the increase in tomato yield per mu ($y$ in kilograms) in the rural complex and the amount of a certain liquid fertilizer used per mu ($x$ in kilograms) are as follows:

| $x$ (kg) | $2$ | $4$ | $5$ | $6$ | $8$ |
|----------|-----|-----|-----|-----|-----|
| $y$ (kg) | $300$ | $400$ | $400$ | $400$ | $500$ |

$(1)$ Based on the data in the table, a linear regression model can be used to fit the relationship between $y$ and $x$. Calculate the correlation coefficient $r$ and explain it (if $|r| > 0.75$, the linear correlation is very high, and a linear regression model can be used for fitting);  
$(2)$ Find the regression equation of $y$ with respect to $x$ and predict the increase in tomato yield when the amount of liquid fertilizer used per mu is $15$ kilograms.

Given: Correlation coefficient formula $r=\frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})({{y_i}-\overline{y}})}}}{{\sqrt{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}\sum_{i=1}^n{{{({{y_i}-\overline{y}})}^2}}}}}}$, reference data: $\sqrt{10}≈3.16$.  
Regression equation $\hat{y}=\hat{b}x+\hat{a}$, where the least squares estimates of the slope and intercept are $\hat{b}=\frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})({{y_i}-\overline{y}})}}}{{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}}}}$, $\hat{a}=\overline{y}-\hat{b}\overline{x}$.

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