
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 data on the monthly income $x_{i}$ (in thousand yuan) and monthly savings $y_{i}$ (in thousand yuan) of the $i$-th household were obtained. The following results were calculated: $\sum\limits_{i=1}^{10}{{{x}_{i}}=80}$ , $\sum\limits_{i=1}^{10}{{{y}_{i}}=}20$, $\sum\limits_{i=1}^{10}{{{x}_{i}}{{y}_{i}}=}184$, $\sum\limits_{i=1}^{10}{x_{i}^{2}=720}$. It is known that the linear regression equation of the monthly savings $y$ with respect to the monthly income $x$ is $\hat{y}=\hat{b}x+\hat{a}$. Determine whether the variables $x$ and $y$ are positively or negatively correlated. If a household in this residential area has a monthly income of $7$ thousand yuan, predict its monthly savings.

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