
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
An enterprise, in response to the national call to gather scientific research forces and strengthen technological innovation, is preparing to increase research and development funding. In order to understand the impact of annual research and development funding $x$ (unit: billion yuan) on annual profit $y$ (unit: billion yuan), an analysis was conducted on the annual research and development funding $x_{i}$ and annual profit $y_{i}$ (for $i=1,2,\ldots,10$) data during the 10-year period of the "Twelfth Five-Year Plan" and the "Thirteenth Five-Year Plan. Two function models were established: $y=\alpha +\beta x^{2}$ and $y=e^{\lambda x+t}$, where $\alpha$, $\beta$, $\lambda$, and $t$ are all constants, and $e$ is the base of the natural logarithm. Let ${u}_{i}={x}_{i}^{2}$ and $v_{i}=\ln y_{i}$. After calculation, the following data were obtained:

| $\overline{x}=26$ | $\overline{y}=215$ | $\overline{u}=680$ | $\overline{v}=5.36$ |
|-------------------|-------------------|------------------|-------------------|
| $\sum_{i=1}^{10}{{({x}_{i}-\overline{x})}^{2}}=100$ | $\sum_{i=1}^{10}{{({u}_{i}-\overline{u})}^{2}}=22500$ | $\sum_{i=1}^{10}{({u}_{i}-\overline{u})}({y}_{i}-\overline{y})=260$ | $\sum_{i=1}^{10}{{({y}_{i}-\overline{y})}^{2}}=4$ |
| $\sum_{i=1}^{10}{{({v}_{i}-\overline{v})}^{2}}=4$ | $\sum_{i=1}^{10}{({x}_{i}-\overline{x})}({v}_{i}-\overline{v})=18$ | | |

$(1)$ From the perspective of the correlation coefficient, analyze which model has a better fit?<br/>
$(2)$ Based on the choice in $(1)$ and the data in the table, establish the regression equation of $y$ on $x$; (coefficients accurate to $0.01$)<br/>
$(3)$ If the profit $y$ in the year 2024 is expected to be 800 billion yuan, predict how much research and development funding $x$ will be in 2024? (result accurate to $0.01$)<br/>
Reference data: $\ln 2=0.693$, $\ln 5=1.609$.<br/>
Regression line: $\hat{y}=\hat{b}x+\hat{a}$, where $\hat{b}=\frac{\sum_{i=1}^{n}({x}_{i}-\overline{x})({y}_{i}-\overline{y})}{\sum_{i=1}^{n}({x}_{i}-\overline{x})^{2}}$ and $\hat{a}=\overline{y}-\hat{b}\overline{x}$.

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