
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 thousand yuan) and monthly savings $y_i$ (in thousand yuan) of the $i$-th household were obtained. The following data was calculated: $\sum_{i=1}^{10} x_i = 80$, $\sum_{i=1}^{10} y_i = 20$, $\sum_{i=1}^{10} x_i y_i = 184$, $\sum_{i=1}^{10} x_i^2 = 720$.

(1) Find the linear regression equation $y = bx + a$ for monthly savings $y$ with respect to monthly income $x$.
(2) Predict the monthly savings for a household in this residential area with a monthly income of $7$ thousand yuan.

(Note: In the linear regression equation $y = bx + a$, $b = \frac{\sum_{i=1}^{n} x_i y_i - n\bar{x}\bar{y}}{\sum_{i=1}^{n} x_i^2 - n\bar{x}^2}$ and $a = \bar{y} - b\bar{x}$, where $\bar{x}$ and $\bar{y}$ are the sample means.)

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