
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 random sample of $10$ households was taken from a residential area, and the monthly income $x_{i}$ (in thousand yuan) and monthly savings $y_{i}$ (in thousand yuan) of the $i$-th household were obtained. The following data were calculated: $\sum_{i=1}^{10}x_{i}=80$, $\sum_{i=1}^{10}y_{i}=20$, $\sum_{i=1}^{10}x_{i}y_{i}=184$, $\sum_{i=1}^{10} x_{ i }^{ 2 }=720$.

1. Find the linear regression equation $\hat{y}=\hat{b}x+\hat{a}$ of the monthly savings $y$ with respect to the monthly income $x$.
2. Predict the monthly savings of a household in this residential area with a monthly income of $7$ thousand yuan.

Note: The formulas for estimating the slope and intercept of the regression line using the least squares method are: $\hat{b}=\frac{\sum_{i=1}^{n}x_{i}y_{i}-n\bar{x}\bar{y}}{\sum_{i=1}^{n}x_{i}^{2}-n\bar{x}^{2}}$ and $\hat{a}=\bar{y}-\hat{b}\bar{x}$.

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