
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 certain chain convenience store sold 2000 types of products from 2014 to 2018. Starting from 2019, the convenience store underwent a comprehensive upgrade and started selling 3000 types of products. The table below shows the profit from 2014 to 2023.

| Year $x$ | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 |
|----------|------|------|------|------|------|------|------|------|------|------|
| Profit $y$ (in $10,000$ yuan) | $27.6$ | $42.0$ | $38.4$ | $48.0$ | $63.6$ | $63.7$ | $72.8$ | $80.1$ | $60.5$ | $99.3$ |

$(1)$ If the profit in a certain year exceeds $45.0$ million yuan, the convenience store will be selected as a model store for that year. If the profit does not exceed $45.0$ million yuan, the convenience store will not be selected as a model store for that year. Complete the $2 \times 2$ contingency table and determine if the number of product types is related to whether the convenience store is selected as a model store (significance level $\alpha = 0.05$, $P(\chi^2 \geq 3.841) \approx 0.05$).

|                   | 2000 types | 3000 types | Total |
|-------------------|------------|------------|-------|
| Selected as model | 2          | 5          | 7     |
| Not selected      | 3          | 0          | 3     |
| Total             | 5          | 5          | 10    |

$(2)$ Based on the data from 2014 to 2023 (excluding 2022), establish a linear regression model $y$ on $x$ (Model 1); based on the data from 2019 to 2023, establish a linear regression model $y$ on $x$ (Model 2). Use these two models to predict the profit of the convenience store in 2024 and explain if these predictions are reliable. (Regression coefficients are accurate to $0.001$, profit is accurate to $0.1$ million yuan).

The formulas for regression coefficients $\hat{a}$ and $\hat{b}$ are as follows:
$\hat{a} = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i - \bar{x})^2} = \frac{\sum_{i=1}^{n}x_i y_i - n\bar{x}\bar{y}}{\sum_{i=1}^{n}x_i^2 - n\bar{x}^2}$,
$\hat{b} = \bar{y} - \hat{a}\bar{x} = \frac{\sum_{i=1}^{n}y_i - \hat{a}\sum_{i=1}^{n}x_i}{n}$.

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