
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
The $5G$ technology is very important to society and the country. From a strategic perspective, the industry generally defines it as the fourth industrial revolution after the steam engine revolution, the electrical revolution, and the computer revolution. A certain technology group produces two core components of $5G$ communication base stations, $A$ and $B$. The table below shows the data of the research and development investment $x$ (in billion yuan) and the revenue $y$ (in billion yuan) of the $A$ component in recent years by the technology group:

| Research Investment $x$ (billion yuan) | 1 | 2 | 3 | 4 | 5 |
|---------------------------------------|---|---|---|---|---|
| Revenue $y$ (billion yuan)             | 3 | 7 | 9 | 10| 11|

$(1)$ Use the sample correlation coefficient $r$ to determine if a linear regression model can be used to fit the relationship between $y$ and $x$ (when $|r| \in [0.75, 1]$, it can be considered that the two variables have a strong linear correlation);  
$(2)$ Find the empirical regression equation of $y$ with respect to $x$. If the revenue from producing the $A$ component is not less than 15 billion yuan, estimate how much research and development funding is needed at least? (Round to $0.001$ billion yuan)  

Given: Sample correlation coefficient $r = \frac{\sum_{i=1}^{n}({x}_{i}-\overline{x})({y}_{i}-\overline{y})}{\sqrt{\sum_{i=1}^{n}{({x}_{i}-\overline{x})}^{2}}\sqrt{\sum_{i=1}^{n}{({y}_{i}-\overline{y})}^{2}}}$, slope of the regression line $\hat{b} = \frac{\sum_{i=1}^{n}({x}_{i}-\overline{x})({y}_{i}-\overline{y})}{\sum_{i=1}^{n}{({x}_{i}-\overline{x})}^{2}}$, intercept $\hat{a} = \overline{y} - \hat{b}\overline{x}$.

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