
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 company in China has developed an environmentally friendly product. To ensure cost-effectiveness, the selling price of each product is not less than $43. After research, the relationship between the selling price $x$ (in yuan per item) and the monthly sales volume $y$ (in ten thousand items) is as shown in the table below:

| Selling Price $x$ (yuan per item) | $52$ | $50$ | $48$ | $45$ | $44$ | $43$ |
|-----------------------------------|------|------|------|------|------|------|
| Monthly Sales Volume $y$ (ten thousand items) | $5$ | $6$ | $7$ | $8$ | $10$ | $12$ |

$(1)$ Calculate the correlation coefficient $r$ (round to two decimal places);  
$(2)$ Establish the empirical regression equation of $y$ with respect to $x$, and estimate the monthly sales volume of the product when the selling price is $55$ yuan per item.

Reference formula: For a set of data $(x_{i}, y_{i})$ $(i=1,2,3,\cdots ,n)$, the correlation coefficient $r$ is calculated as follows: $r=\frac{\sum_{i=1}^{n}{({x}_{i}-\overline{x})}({y}_{i}-\overline{y})}{\sqrt{\sum_{i=1}^{n}{{({x}_{i}-\overline{x})}^{2}}}\sqrt{\sum_{i=1}^{n}{{({y}_{i}-\overline{y})}^{2}}}$, and the slope and intercept of the regression line $\hat{y}=\hat{b}x+\hat{a}$ are estimated by the least squares method as follows: $\hat{b}=\frac{\sum_{i=1}^{n}({x}_{i}-\overline{x})({y}_{i}-\overline{y})}{\sum_{i=1}^{n}{({x}_{i}-\overline{x})}^{2}},\hat{a}=\overline{y}-\hat{b}•\overline{x}．(\sqrt{34}≈5.83)$

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