
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 to understand the impact of annual research and development funding $x$ (in units of billion yuan) on annual profit $y$ (in units of billion yuan). By analyzing the data of annual research and development funding $x_i$ and annual profit $y_i$ ($i=1,2,\cdots ,10$) during the 10-year periods of the "Twelfth Five-Year Plan" and "Thirteenth Five-Year Plan," two function models were established: $y=\alpha +\beta x^{2}$; $y=e^{\lambda x+t}$, where $\alpha$, $\beta$, $\lambda$, $t$ are constants, $e$ is the base of natural logarithm. Let ${u}_{i}={x}_{i}^{2}$, $v_{i}=\ln y_{i}$ ($i=1,2,\cdots ,10$). After calculation, the following data were obtained:<br/><table><tbody><tr><td align="middle">$\overline{x}=26$</td><td align="middle">$\overline{y}=215$</td><td align="middle">$\overline{u}=680$</td><td align="middle">$\overline{v}=5.36$</td></tr><tr><td align="middle">$\sum_{i=1}^{10}{{({x}_{i}-\overline{x})}^{2}}=100$</td><td align="middle">$\sum_{i=1}^{10}{{({u}_{i}-\overline{u})}^{2}}=22500$</td><td align="middle">$\sum_{i=1}^{10}{({u}_{i}-\overline{u})}({y}_{i}-\overline{y})=260$</td><td align="middle">$\sum_{i=1}^{10}{{({y}_{i}-\overline{y})}^{2}}=4$</td></tr><tr><td align="middle">$\sum_{i=1}^{10}{{({v}_{i}-\overline{v})}^{2}}=4$</td><td align="middle">$\sum_{i=1}^{10}{({x}_{i}-\overline{x})}({v}_{i}-\overline{v})=18$</td><td align="middle"></td><td align="middle"></td></tr></tbody></table>$(1)$ 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/>Attachment: Correlation coefficient $r=\frac{\sum_{i=1}^{n}{({x}_{i}-\overline{x})}({y}_{i}-\overline{y})}{\sqrt{\sum_{i=1}^{n}{{({x}_{i}-\overline{x})}^{2}}}\sqrt{\sum_{i=1}^{n}{{({y}_{i}-\overline{y})}^{2}}}$<br/>Regression line $\hat{y}=\hat{b}x+\hat{a}$: $\hat{b}=\frac{\sum_{i=1}^{n}{({x}_{i}-\overline{x})}({y}_{i}-\overline{y})}{\sum_{i=1}^{n}{{({x}_{i}-\overline{x})}^{2}}}$, $\hat{a}=\overline{y}-\hat{b}\overline{x}$.

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