
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
In early January 2022, a new type of respiratory infectious disease broke out in a city. This disease is highly contagious. In order to quickly control the epidemic caused by this infectious disease, the epidemic monitoring agency of the city has compiled the data on the number of new cases added each day from January 12th to January 15th. The statistical data is as follows:

| January $x$ | 12 | 13 | 14 | 15 |
|-------------|----|----|----|----|
| New Cases $y$ | 26 | 29 | 28 | 31 |

$(1)$ The epidemic monitoring agency conducted a linear regression analysis on the statistical data above to establish a linear relationship between $y$ and $x$ based on the data in the table. Find the linear regression equation of $y$ with respect to $x$, $\hat{y} = \hat{b}x + \hat{a}$.

$(2)$ Predict on which day the number of new cases will exceed 36 people.

Given: For a set of data points $(x_{1}, y_{1}), (x_{2}, y_{2}), \ldots, (x_{n}, y_{n})$, the least squares estimates of the slope and intercept of the regression equation $\hat{y} = \hat{b}x + \hat{a}$ are: $\hat{b} = \frac{\sum_{i=1}^{n}x_{i}y_{i} - n\overline{xy}}{\sum_{i=1}^{n}x_{i}^{2} - n\overline{x}^{2}} = \frac{\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y})}{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}$, $\hat{a} = \overline{y} - \hat{b}\overline{x}$. Reference data: $\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y}) = 7$.

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