
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
Research shows that if the temperature difference is too large, people may not pay attention to keeping warm, which may lead to self-exposure to cold stimulation, increasing the probability of catching a cold, especially for children and the elderly. A math modeling team from a certain high school studied the relationship between the size of the day-night temperature difference and the number of new students visiting the school clinic due to a cold at a primary school. They recorded the temperature difference for six days and checked the number of new student visits for a cold each day in the school clinic, obtaining the following data:

| Date | Day 1 | Day 2 | Day 3 | Day 4 | Day 5 | Day 6 |
|------|-------|-------|-------|-------|-------|-------|
| Day-night temperature difference $x(^{\circ}C)$ | 4 | 7 | 8 | 9 | 14 | 12 |
| New student visits for a cold $y$(people) | $y_{1}$ | $y_{2}$ | $y_{3}$ | $y_{4}$ | $y_{5}$ | $y_{6}$ |

Reference data: $\sum_{i=1}^6{y_i^2=3463}$, $\sum_{i=1}^6{{{({{y_i}-y})}^2}=289}$.

$(1)$ Given that there are 4 boys among the new students visiting the clinic on the first day, and 2 students are randomly selected from the new students on the first day, with the number of boys denoted as $X$. If the probability of selecting at least one girl from the 2 students is $\frac{5}{6}$, find the probability distribution and mathematical expectation of the random variable $X$.

$(2)$ Given that the sample correlation coefficient between two variables $x$ and $y$ is $r=\frac{{16}}{{17}}$, use the method of least squares to find the empirical regression equation of $y$ on $x$, $\hat{y}=\hat{b}x+\hat{a}$. Based on this, estimate the number of new students visiting the school clinic due to a cold when the day-night temperature difference is $15^{\circ}C$.

Reference formulas: $\hat{b}=\frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})}({{y_i}-\overline{y}})}}{{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}}}}$, $r=\frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})}({{y_i}-\overline{y}})}}{{\sqrt{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}}}⋅\sqrt{\sum_{i=1}^n{{{({{y_i}-\overline{y}})}^2}}}}}$.

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