
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
Small household appliances refer to household appliances other than high-power and large-volume household appliances (such as refrigerators, washing machines, air conditioners, etc.). With a wide range of applications, the market for smart small household appliances has been continuously growing in recent years. The table below shows the market size of the smart small household appliances market in China for 5 consecutive years (unit: billion yuan), with the corresponding codes for each year being 1 to 5.

| Year Code $x$ | 1    | 2    | 3    | 4    | 5    |
|--------------|------|------|------|------|------|
| Market Size $y$ | 0.9  | 1.2  | 1.5  | 1.4  | 1.6  |

$(1)$ Based on the data in the table, a linear regression model can be used to fit the relationship between $y$ and $x$. Please explain using the correlation coefficient.  
$(2)$ Establish an empirical regression equation for $y$ in terms of $x$ (coefficients accurate to $0.01$).  

Reference data: $\overline{y}=1.32, \sum_{i=1}^5{x_i y_i}=21.4, \sqrt{\sum_{i=1}^5{(y_i-\overline{y})^2}} \approx 0.55, \sqrt{10} \approx 3.16$;  
Reference formulas: Correlation coefficient $r=\frac{\sum_{i=1}^n{(x_i-\overline{x})(y_i-\overline{y})}}{\sqrt{\sum_{i=1}^n{(x_i-\overline{x})^2}}\sqrt{\sum_{i=1}^n{(y_i-\overline{y})^2}}}$, regression equation $\hat{y}=\hat{b}x+\hat{a}$ where the least squares estimates of the slope and intercept are $\hat{b}=\frac{\sum_{i=1}^n{(x_i-\overline{x})(y_i-\overline{y})}}{\sum_{i=1}^n{(x_i-\overline{x})^2}}, \hat{a}=\overline{y}-\hat{b}\overline{x}$.

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