
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
From a randomly selected sample of 10 families in a residential area, we obtained data on the monthly income $x_i$ (in thousands of yuan) and monthly savings $y_i$ (in thousands of yuan) for the $i$-th family. It was calculated that $\sum\limits_{i=1}^{10}{x_i}=80$, $\sum\limits_{i=1}^{10}{y_i}=20$, $\sum\limits_{i=1}^{10}{x_iy_i}=184$, and $\sum\limits_{i=1}^{10}{{x_i}^2}=720$.
(Ⅰ) Find the linear regression equation of the family's monthly savings $y$ on monthly income $x$, $\bar{y}= \bar{b}x+ \bar{a}$;
(Ⅱ) Determine whether the variables $x$ and $y$ are positively or negatively correlated;
(Ⅲ) If a family in this residential area has a monthly income of 12 thousand yuan, predict the family's monthly savings.
Note: In the linear regression equation $\bar{y}= \bar{b}x+ \bar{a}$, $b= \frac{ \sum\nolimits_{i=1}^{n}{x_iy_i-n \bar{x \bar{y}}}}{ \sum\nolimits_{i=1}^{n}{{x_i}^2}-n{ \bar{x}}^2}$, $\bar{a}= \bar{y}- \bar{b} \bar{x}$. Where $\bar{x}, \bar{y}$ are the sample means, the linear regression equation can also be written as $\land y=\land bx+\land a$.

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