
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 thousands of yuan) and monthly savings $y\_i$ (in thousands of yuan) of the $i$th household were obtained. The following data were calculated: $\sum\_{i=1}^{10}{{{x}\_{i}}}=80$, $\sum\_{i=1}^{10}{{{y}\_{i}}}=20$, $\sum\_{i=1}^{10}{{{x}\_{i}}{{y}\_{i}}}=184$, $\sum\_{i=1}^{10}{x\_{i}^{2}=720}$.

(I) Find the linear regression equation $\hat{y}=\hat{b}x+\hat{a}$ of monthly savings $y$ on monthly income $x$.

(II) Determine whether the variables $x$ and $y$ are positively or negatively correlated.

(III) Predict the monthly savings of a household in this residential area with a monthly income of $12$ thousand yuan.

Note: In the linear regression equation $\hat{y}=\hat{b}x+\hat{a}$, $\hat{b}=\dfrac{\sum\_{i=1}^{n}{({{x}\_{i}}-\bar{x})({{y}\_{i}}-\bar{y})}}{\sum\_{i=1}^{n}{{{({{x}\_{i}}-\bar{x})}^{2}}}}=\dfrac{\sum\_{i=1}^{n}{{{x}\_{i}}{{y}\_{i}}}-n\bar{x}\bar{y}}{\sum\_{i=1}^{n}{x\_{i}^{2}}-n{{\bar{x}}^{2}}}$, and $\hat{a}=\bar{y}-\hat{b}\bar{x}$, where $\bar{x}$ and $\bar{y}$ are the sample means.

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