
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
After many years of environmental management in a certain area, the barren mountains have been transformed into green waters and green mountains. To estimate the total volume of a certain type of tree in a forest area, $10$ trees of this type were randomly selected, and the root cross-sectional area (unit: $m^{2}$) and volume (unit: $m^{3}$) of each tree were measured, resulting in the following data:<br/>

| Sample $i$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | Sum |
|------------|-----|-----|-----|-----|-----|-----|-----|-----|-----|------|-----|
| Root Cross-sectional Area $x_{i}$ | $0.04$ | $0.06$ | $0.04$ | $0.08$ | $0.08$ | $0.05$ | $0.05$ | $0.07$ | $0.07$ | $0.06$ | $0.6$ |
| Volume $y_{i}$ | $0.25$ | $0.40$ | $0.22$ | $0.54$ | $0.51$ | $0.34$ | $0.36$ | $0.46$ | $0.42$ | $0.40$ | $3.9$ |

It is also given that $\sum_{i=1}^{10}x_{i}^{2}=0.038$, $\sum_{i=1}^{10}y_{i}^{2}=1.6158$, and $\sum_{i=1}^{10}x_{i}y_{i}=0.2474$.<br/>

$(1)$ Estimate the average root cross-sectional area and volume of this type of tree in the forest area;<br/>
$(2)$ Calculate the sample correlation coefficient between the root cross-sectional area and volume of this type of tree in the forest area (round to $0.01$);<br/>
$(3)$ Now, the root cross-sectional area of all trees of this type in the forest area has been measured, and the total root cross-sectional area of all trees of this type is $186m^{2}$. It is known that the volume of the trees is approximately proportional to their root cross-sectional area. Use the given data to provide an estimate of the total volume of this type of tree in the forest area.<br/>

Given: $r=\frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})}({{y_i}-\overline{y}})}}{{\sqrt{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}}\sum_{i=1}^n{{{({{y_i}-\overline{y}})}^2}}}}}$, $\sqrt{1.896}\approx 1.377$.

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