
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 units of thousands of yuan) and monthly savings $y_{i}$ (in units of thousands of yuan) for each household were obtained. The following data was calculated: $\sum\limits_{i=1}^{10}x_{i}=80$, $\sum\limits_{i=1}^{10}y_{i}=20$, $\sum\limits_{i=1}^{10}x_{i}y_{i}=184$, $\sum\limits_{i=1}^{10}x_{i}^{2}=720$.
(I) 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.
(II) Predict the monthly savings for a household in this residential area with a monthly income of $7,000$ yuan.

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

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