
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 drawn from a residential area, and data on 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. It was calculated that $\sum\limits_{i=1}^{10}{{{x}_{i}}}=80$, $\sum\limits_{i=1}^{10}{{{y}_{i}}}=20$, $\sum\limits_{i=1}^{10}{{{x}_{i}}{{y}_{i}}}=184$, $\sum\limits_{i=1}^{10}{x_{i}^{2}}=720$.

(1) Find the linear regression equation $y=bx+a$ of the monthly savings $y$ on the monthly income $x$,

(2) Determine whether the variables $x$ and $y$ are positively or negatively correlated,

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

Note: In the linear regression equation $y=bx+a$, $b=\frac{\sum\limits_{i=1}^{n}{{{x}_{i}}{{y}_{i}}-n\bar{x}\cdot \bar{y}}}{\sum\limits_{i=1}^{n}{x_{i}^{2}-n{{{\bar{x}}}^{2}}}}$, $a=\bar{y}-b\bar{x}$, where $\bar{x}$, $\bar{y}$ are the sample means, and the linear regression equation can also be written as $\widehat{y}=\widehat{b}x+\widehat{a}$.

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