
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, a barren mountain in a certain area has been transformed into green hills and clear waters. To estimate the total volume of a certain type of tree in a forest area, $10$ trees of this type were randomly selected. 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 $i$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ | Average |
|------------|-----|-----|-----|-----|-----|-----|-----|-----|-----|------|---------|
| Root Cross-sectional Area $x_{i}$ | $0.04$ | $0.06$ | $0.04$ | $0.08$ | $0.08$ | $0.05$ | $a$ | $b$ | $c$ | $0.07$ | $0.06$ |
| Volume $y_{i}$ | $0.25$ | $0.41$ | $0.22$ | $0.54$ | $0.53$ | $0.34$ | $0.35$ | $0.39$ | $0.43$ | $0.44$ | $0.39$ |

where $a$, $b$, and $c$ form an arithmetic sequence, and $\sum_{i=1}^{6}{{x_i}{y_i}}=0.146$, $\sqrt{\sum_{i=1}^{6}{x_i^2}-6{{\overline{x}}^2}}\approx0.044$, $\sqrt{\sum_{i=1}^{6}{y_i^2}-6{{\overline{y}}^2}}\approx0.303$.<br/>$(1)$ Find the value of $b$;<br/>$(2)$ If the data corresponding to the first $6$ sample numbers are selected, determine whether there is a strong linear correlation between the root cross-sectional area and the volume of this type of tree in the forest area, and find the regression line equation for the root cross-sectional area and volume of this type of tree in the forest area $(0.25\leqslant |r|\leqslant 0.75$, the correlation between the two variables is considered moderate; if $|r| \gt 0.75$, the correlation between the two variables is considered strong);<br/>Note: The correlation coefficient $r=\frac{{\sum_{i=1}^n{{x_i}{y_i}}-n\overline{x}\overline{y}}}{{\sqrt{({\sum_{i=1}^n{x_i^2}-n{{\overline{x}}^2}})({\sum_{i=1}^n{y_i^2}-n{{\overline{y}}^2}})}}}$,<br/>In the regression line $y=bx+\hat{a}$, $\hat{b}=\frac{{\sum_{i=1}^n{{x_i}{y_i}}-n\overline{x}\overline{y}}}{{\sum_{i=1}^n{x_i^2}-n{{\overline{x}}^2}}}$, $\hat{a}=\overline{y}-\hat{b}\overline{x}$.<br/>$(3)$ Estimate the values of $a$ and $c$ based on the regression line equation (rounded to $0.01$).

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