
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 high school organized students to investigate and analyze the monthly profit situation of a certain physical store for some months in the year $2022$ offline. The data obtained is as follows:

| Month $x$ | $2$ | $4$ | $6$ | $8$ | $10$ | $12$ |
|-----------|-----|-----|-----|-----|------|------|
| Net Profit $y$ (in $10,000$ yuan) | $0.9$ | $2.0$ | $4.2$ | $3.9$ | $5.2$ | $5.1$ |
| $\lambda =\ln x$ | $0.7$ | $1.4$ | $1.8$ | $2.1$ | $2.3$ | $2.5$ |
| $\mu=\sqrt{x}$ | $1.4$ | $2.0$ | $2.4$ | $2.8$ | $3.2$ | $3.5$ |

Based on the scatter plot, it is planned to establish the regression equation of $y$ on $x$ using either model ① $y=a\ln x+b$ or model ② $y=c\sqrt{x}+d$.

$(1)$ Using the linear correlation coefficient, determine which of the two models above is more suitable as the regression equation of $y$ on $x.

$(2)$ Based on the reference data, according to the result of $(1)$, find the regression equation of $y$ on $x (rounded to $0.1)$.

Given data: $\overline{x}=7$, $\overline{y}=3.55$, $\overline \lambda=1.80$, $\overline \mu=2.55$, $\sum_{i=1}^6{(\lambda_i-\overline \lambda)^2}=2.20$, $\sum_{i=1}^6{(\mu_i-\overline \mu)^2}=2.89$, $\sum_{i=1}^6(y_i-\overline{y})(\lambda_i-\overline \lambda)=5.55$, $\sum_{i=1}^6(y_i-\overline{y})(\mu_i-\overline \mu)=6.32$, $\sqrt{\sum_{i=1}^6{(y_i-\overline{y})^2}\sum_{i=1}^6{(\lambda_i-\overline \lambda)^2}}=5.76$, $\sqrt{\sum_{i=1}^6{(y_i-\overline{y})^2}\sum_{i=1}^6{(\mu_i-\overline \mu)^2}}=6.61$.

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