
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
A certain chip company needs to understand the impact of annual R&D funding $x$ (unit: billion yuan) on annual sales $y$ (unit: billion yuan) in order to formulate next year's R&D investment plan. The company conducted a comparative analysis of historical data and established two function models: ① $y=\alpha +\beta x^{2}$, ② $y=e^{\lambda x+t}$, where $\alpha$, $\beta$, $\lambda$, $t$ are all constants, and $e$ is the base of natural logarithm. The company has processed the data of the annual R&D funding amount $x_{i}$ and annual sales $y_{i}$ for the past 12 years, and obtained the following data after calculating: 

| $\overline{x}$ | $\overline{y}$ | $\sum_{i=1}^{12}{({x}_{i}-\overline{x})}^{2}$ | $\sum_{i=1}^{12}{({y}_{i}-\overline{y})}^{2}$ | $\overline{u}$ | $\overline{v}$ |
|----------------|----------------|------------------------------------------|------------------------------------------|--------------|--------------|
| 20             | 66             | 770                                      | 200                                      | 460          | 4.2          |

| $\sum_{i=1}^{12}{({u}_{i}-\overline{u})}^{2}$ | $\sum_{i=1}^{12}({u}_{i}-\overline{u})({y}_{i}-\overline{y})$ | $\sum_{i=1}^{12}{({v}_{i}-\overline{v})}^{2}$ | $\sum_{i=1}^{12}({x}_{i}-\overline{x})({v}_{i}-\overline{v})$ |
|------------------------------------------|------------------------------------------------------|------------------------------------------|------------------------------------------------------|
| 3125000                                 | 21500                                                | 0.308                                    | 14                                                   |

$(1)$ Let the sample correlation coefficient between $u$ and $y$ be $r_{1}$, and the sample correlation coefficient between $x$ and $v$ be $r_{2}$. From the perspective of the sample correlation coefficient (rounded to $0.01$), determine which model has a better fitting effect. 

$(2)(i)$ Based on the choice in part (1) and the data in the table, establish a nonlinear empirical regression equation of $y$ with respect to $x$. 

$(ii)$ If the sales target for the next year is $90$ billion yuan, approximately how much R&D funding $x$ is needed for the next year? 

Reference data: $308=4\times 77$, $\sqrt{90}≈9.4868$, $e^{4.4998}\approx 90$. 

The correlation coefficient $r$ is calculated as: 

$r=\frac{\sum_{i=1}^{n}{({x}_{i}-\overline{x})}({y}_{i}-\overline{y})}{\sqrt{\sum_{i=1}^{n}{{({x}_{i}-\overline{x})}^{2}}}•\sqrt{\sum_{i=1}^{n}{{({y}_{i}-\overline{y})}^{2}}}}=\frac{\sum_{i=1}^{n}{{x}_{i}}{y}_{i}-n\overline{x}\overline{y}}{\sqrt{\sum_{i=1}^{n}{{x}_{i}^{2}}-n{\overline{x}}^{2}}•\sqrt{\sum_{i=1}^{n}{{y}_{i}^{2}}-n{\overline{y}}^{2}}$.

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