
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 geological exploration team set up one origin point to detect the distribution of lithium metal in a mine. They then determined five sampling points, with the distances from these five sampling points to the origin point denoted as $x_{i}$, where $x_{i}=i$ for $i=1,2,3,4,5$. The team also obtained the lithium content $y_{i}$ at each sampling point, resulting in a set of data $(x_{i}, y_{i})$ for $i=1,2,3,4,5$. After calculations, the team obtained the following statistical values: $\sum_{i=1}^{5}y_{i}=62$, $\sum_{i=1}^{5}(x_{i}-\overline{x})(y_{i}-\overline{y})=47$, $\sum_{i=1}^{5}u_{i}\approx4.79$, $\sum_{i=1}^{5}(u_{i}-\overline{u})^{2}\approx1.615$, and $\sum_{i=1}^{5}(u_{i}-\overline{u})(y_{i}-\overline{y})\approx19.38$, where $u_{i}=\ln x_{i}$ for $i=1,2,3,4,5$.

$(1)$ Use the correlation coefficient to determine which regression model, $y=a+bx$ or $y=a+b\ln x$, is more suitable for modeling $y$ in terms of $x$.

$(2)$ Establish the regression equation for $y$ in terms of $x.

Given formulas: In the regression equation $y=a+bt$, the formulas for the least squares estimates of the slope and intercept, and the correlation coefficient are $b=\frac{\sum_{i=1}^{n}(t_{i}-\overline{t})(y_{i}-\overline{y})}{\sum_{i=1}^{n}(t_{i}-\overline{t})^{2}}=\frac{\sum_{i=1}^{n}t_{i}y_{i}-n\overline{t}y}{\sum_{i=1}^{n}t_{i}^{2}-nt^{2}}$, $a=\overline{y}-b\overline{t}$, and $r=\frac{\sum_{i=1}^{n}(t_{i}-\overline{t})(y_{i}-\overline{y})}{\sqrt{\sum_{i=1}^{n}(t_{i}-\overline{t})^{2}}\sqrt{\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}}}$.

Given data: $\frac{19.38^{2}}{1.615}=232.56$.

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