
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 data ${x_i}$ (in units of thousand yuan) of the $i$-th household was obtained. The following statistical values were calculated: $\sum_{i=1}^{10} x_i = 80$, $\sum_{i=1}^{10} y_i = 20$, $\sum_{i=1}^{10} x_i y_i = 184$, and $\sum_{i=1}^{10} x_i^2 = 720$.

(1) Find the linear regression equation $\hat{y} = \hat{b}x + \hat{a}$ of the monthly savings $y$ with respect to the monthly income $x$, and determine whether the variables $x$ and $y$ are positively or negatively correlated.

(2) Predict the monthly savings of a household in this residential area with a monthly income of $7$ thousand yuan.

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