
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
Online shopping is an important way of shopping for modern young people. As of December 2021, the scale of online shopping users in China has reached 8.42 billion, an increase of 59.68 million compared to December 2020, accounting for 81.6% of the overall netizens. A certain e-commerce platform has been tracking the annual profit $y_i$ (in units of ten thousand yuan) of one of its specialty stores over the past five years, with corresponding time $t_i$ in years as shown in the table below:

| $t_i$ | 1 | 2 | 3 | 4 | 5 |
|-------|---|---|---|---|---|
| $y_i$ | 2.6 | 3.1 | 4.5 | 6.8 | 8.0 |

$(1)$ Based on the data in the table, can a linear regression model be used to fit the relationship between $y$ and $t$? Calculate the correlation coefficient $r$ and explain (calculate the result accurate to $0.01$). (If $|r| \geq 0.75$, the linear correlation is very high and a linear regression model can be used for fitting.)  
$(2)$ Use the least squares method to find the regression equation for profit $y$ and time $t$, and predict the profit when $t=7$.  
Given: $r=\frac{{\sum_{i=1}^n{({{t_i}-\overline{t}})}({{y_i}-\overline{y}})}}{{\sqrt{\sum_{i=1}^n{{{({{t_i}-\overline{t}})}^2}}}\sqrt{\sum_{i=1}^n{{{({{y_i}-\overline{y}})}^2}}}}=\frac{{\sum_{i=1}^n{{t_i}}{y_i}-n\overline{t}\overline{y}}}{{\sqrt{\sum_{i=1}^n{{{({{t_i}-\overline{t}})}^2}}}\sqrt{\sum_{i=1}^n{{{({{y_i}-\overline{y}})}^2}}}}$, $\hat{b}=\frac{{\sum_{i=1}^n{({{t_i}-\overline{t}})}({{y_i}-\overline{y}})}}{{\sum_{i=1}^n{{{({{t_i}-\overline{t}})}^2}}}}=\frac{{\sum_{i=1}^n{{t_i}}{y_i}-n\overline{t}\overline{y}}}{{\sum_{i=1}^n{t_i^2}-n{{\overline{t}}^2}}}$, $\hat{a}=\overline{y}-\hat{b}\overline{t}$.  
Reference data: $\sum_{i=1}^5{{t_i}}{y_i}=89.5$, $\sqrt{\sum_{i=1}^5{{{({{t_i}-\overline{t}})}^2}}}= \sqrt{10}$, $\sqrt{\sum_{i=1}^5{{{({{y_i}-\overline{y}})}^2}}}= \sqrt{21.86}$, $\sqrt{218.6} \approx 14.785$.

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