
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
According to statistics, the annual income of residents in a city (the total income of all residents in a year, in billion yuan) and the sales of a certain type of goods (in billion yuan) over a period of $10$ years are shown in the table below:

| Year $n$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|----------|-----|-----|-----|-----|-----|-----|-----|-----|-----|------|
| Resident Income $x$ | $32.2$ | $31.1$ | $32.9$ | $35.7$ | $37.1$ | $38.0$ | $39.0$ | $43.0$ | $44.6$ | $46.0$ |
| Goods Sales $y$ | $25.0$ | $30.0$ | $34.0$ | $37.0$ | $39.0$ | $41.0$ | $42.0$ | $44.0$ | $48.0$ | $51.0$ |

Based on the table data, the following values of some statistics are obtained:

| $\sum_{i=1}^{10}{x}_{i}$ | $\sum_{i=1}^{10}{y}_{i}$ | $\sum_{10}^{i=1}(x_{i}-\overline{x})^{2}$ | $\sum_{10}^{i=1}(y_{i}-\overline{y})^{2}$ | $\sum_{10}^{i=1}(x_{i}-\overline{x})(y_{i}-\overline{y})$ |
|--------------------------|--------------------------|--------------------------------------------|--------------------------------------------|------------------------------------------------------------|
| $379.6$                  | $391$                    | $246.904$                                  | $568.9$                                    | $m$                                                        |

$(1)$ Based on the data in the table, the sample correlation coefficient $r\approx 0.95$. Can it be inferred that the linear correlation between $y$ and $x$ is very strong?  
$(2)$ Based on the values of the statistics and the sample correlation coefficient $r\approx 0.95$, establish the empirical regression equation of $y$ on $x$ (coefficients accurate to $0.01$);  
$(3)$ Based on the regression equation in $(2)$, calculate the residual corresponding to the first sample point $\left(32.2, 25.0\right)$ (accurate to $0.01$); and determine whether the value of $\hat{b}$ will increase or decrease if this sample point is excluded from the regression analysis (no need to provide a reason, just make a judgment).  
Attachment: For a sample $(x_{i}, y_{i}) \left(i=1,2,\cdots ,n\right)$, the correlation coefficient $r=\frac{\sum_{i=1}^{n}{({x}_{i}-\overline{x})}({y}_{i}-\overline{y})}{\sqrt{\sum_{i=1}^{n}{{({x}_{i}-\overline{x})}^{2}}}\sqrt{\sum_{i=1}^{n}{{({y}_{i}-\overline{y})}^{2}}}}$, $\sqrt{2.297}\approx 1.516$, $\hat{b}=\frac{\sum_{i=1}^{n}{({x}_{i}-\overline{x})}({y}_{i}-\overline{y})}{\sum_{i=1}^{n}{{({x}_{i}-\overline{x})}^{2}}}$, $\hat{a}=\overline{y}-\hat{b}\overline{x}$.

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