
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
The data between the protein content $x(\%)$ of a certain dairy product produced by a company and the production cost $y (\text{元})$ are as follows:

|x|0|0.69|1.39|1.79|2.40|2.56|2.94|
|-|-|-|-|-|-|-|-|
|y|19|32|40|44|52|53|54|

It is known that there is a linear relationship between the production cost $y$ and the protein content $x$ of the product.

$(1)$ Find the regression equation of the production cost $y$ with respect to the protein content $x$.

$(2)$ Based on the result of $(1)$, if the company plans to increase the production cost to $60$ to $70$ yuan, determine the range of protein content values of the dairy product. (Round to two decimal places)

Reference formula: $\hat{b}=\frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})({{y_i}-\overline{y}})}}}{{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}}}}$.

Reference data: $\overline{x}=1.68$, $\sum_{i=1}^7{{{({{x_i}-\overline{x}})}^2}=6.79}$, $\sum_{i=1}^7{({{x_i}-\overline{x}})({{y_i}-\overline{y}})=81.41}$.

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