
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]

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[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

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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 linear regression equation for $y$ with respect to $x$ and predict the increase in tomato yield when the amount of liquid fertilizer used per mu is 15 kilograms.

Given: For a set of data $(x_{1}, y_{1}), (x_{2}, y_{2}), \ldots, (x_{n}, y_{n})$, the least squares estimates for the slope and intercept of the regression line equation $\hat{y} = \hat{b}x + \hat{a}$ are $\hat{b} = \frac{\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})}{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}$ and $\hat{a} = \overline{y} - \hat{b}\overline{x}$, and the correlation coefficient $r = \frac{\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})}{\sqrt{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\sqrt{\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}}}$.

Reference data: $\sqrt{10} \approx 3.16$.

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