
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
To help farmers increase their income, a certain region has adopted live streaming to promote the sales of agricultural and sideline products, utilizing the credibility of mainstream media and aggregating the power of sales anchors to help streamline the agricultural product supply chain. The table below shows the income situation of village cadres in the region from January to May 2023 through live streaming sales of agricultural and sideline products. It is known that the relationship between the month $(t)$ and the income $(y)$ can be fitted using a linear regression model.

| Month $(t)$ | 1   | 2   | 3   | 4   | 5   |
|-------------|-----|-----|-----|-----|-----|
| Income $(y)$ (in ten thousand yuan) | 0.3 | 0.3 | 0.5 | 0.9 | 1   |

$(1)$ Find the regression line equation for $y$ in terms of $t$.

$(2)$ Determine whether the income from agricultural and sideline products in September in this region can exceed 20,000 yuan, and explain the reason.

Reference formula: $\hat{b}=\frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})}({{y_i}-\overline{y}})}}{{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}}}}=\frac{{\sum_{i=1}^n{{x_i}}{y_i}-n\overline{x}\;\overline{y}}}{{\sum_{i=1}^n{x_i^2}-n{{\overline{x}}^2}}}$, $\hat{a}=\overline{y}-\hat{b}\overline{x}$.

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