
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]


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In recent years, the international environment and situation have become increasingly severe, with fierce competition in high-tech industries. The country has called on various high-tech enterprises to gather research forces, strengthen technological innovation, and significantly increase research and development funds to break through the "bottleneck" key technologies in various fields in China. In order to understand the research investment and output of high-tech enterprises in the city, a certain city randomly selected 8 semiconductor companies and collected data on their research investment $x$ (in hundred billion yuan) and the additional income $y$ (in hundred billion yuan) generated from this investment from 2018 to 2022. It was found that the investment $x$ and the additional income $y$ are linearly related.

Given the following data:

| Research Investment $x_i$ (in hundred billion yuan) | 2 | 3 | 4 | 5 | 6 | 8 | 9 | 11 |
|-----------------------------------------------------|---|---|---|---|---|---|---|----|
| Additional Income $y_i$ (in hundred billion yuan)   | 3.6 | 4.1 | 4.8 | 5.4 | 6.2 | 7.5 | 7.9 | 9.1 |

$(1)$ Find the linear regression equation between the additional income $y$ and the research investment $x$ (round to three decimal places);  
$(2)$ Now, from these 8 companies with an investment of at least 5 hundred billion yuan, randomly select 3 companies. Find the probability that exactly 1 of the 3 selected companies has additional income greater than the investment.

Given reference data: $\sum_{i=1}^8{x_i}{y_i}=334.1$, $\sum_{i=1}^8{y_i}=48.6$, $\sum_{i=1}^8{x_i^2}=356$.

In the linear regression equation $\hat{y}=\hat{b}x+\hat{a}$, $\hat{b}=\frac{\sum_{i=1}^n{(x_i-\overline{x})}(y_i-\overline{y})}{\sum_{i=1}^n{(x_i-\overline{x})^2}}=\frac{\sum_{i=1}^n{x_i}{y_i}-n\overline{x}\cdot\overline{y}}{\sum_{i=1}^n{x_i^2}-n\overline{x}^2}$, $\hat{a}=\overline{y}-\hat{b}\overline{x}$.

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