
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 table below shows the average family income per capita (in units of 10,000 yuan) of rural residents from 2018 to 2022.

| Year | 2018 | 2019 | 2020 | 2021 | 2022 |
|------|------|------|------|------|------|
| Year Code $x$ | 1 | 2 | 3 | 4 | 5 |
| Average Income $y$ (10,000 yuan) | 1.2 | 1.4 | 1.5 | 1.6 | 1.8 |

$(1)$ Use the correlation coefficient $r$ to determine the strength of the relationship between $y$ and $x$ (when $0.75 < |r| \leq 1$, the relationship between $y$ and $x$ is strong; otherwise, it is weak, accurate to $0.01$). $(2)$ Find the linear regression equation of $y$ with respect to $x$, $\hat{y} = \hat{b}x + \hat{a}$, and predict the average family income of rural residents in 2023. Note: For a set of data $(x_{1}, y_{1}), (x_{2}, y_{2}), \ldots, (x_{n}, y_{n})$, the least squares estimates of the slope and intercept of the regression line $\hat{y} = \hat{b}x + \hat{a}$ are $\hat{b} = \frac{\sum_{i=1}^{n}x_{i} \cdot y_{i} - n \cdot \overline{x} \cdot \overline{y}}{\sum_{i=1}^{n}x_{i}^{2} - n \cdot \overline{x}^{2}}$ and $\hat{a} = \overline{y} - \hat{b} \overline{x}$, where the sample correlation coefficient $r = \frac{\sum_{i=1}^{n}x_{i} \cdot y_{i} - n \cdot \overline{x} \cdot \overline{y}}{\sqrt{\sum_{i=1}^{n}(x_{i} - \overline{x})^{2}} \cdot \sqrt{\sum_{i=1}^{n}(y_{i} - \overline{y})^{2}}}$. Reference data: $\sqrt{2} \approx 1.414$.

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