
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
Recently, the issue of "10-minute break" for primary and secondary school students has received widespread attention in some places. The country has called for an increase in outdoor activity time for students. However, after entering December, the weather gradually becomes colder, and many students have reduced the number of outdoor activities due to low temperatures. To understand the situation of this class, a student has collected data on the temperature changes and the number of students going out of the building during a fixed break time for this class for a week (5 days), and obtained the following data:

| Temperature $x_{i}$ ($^\circ\mathrm{C}$) | 7 | 10 | 11 | 15 | 17 |
|-----------------------------------------|---|----|----|----|----|
| Number of students going out $y_{i}$    | 20| 16 | 17 | 10 | 7  |

$(1)$ Using the method of least squares, find the linear regression equation between variables $x$ and $y$.

Given: The coefficients of the linear regression equation $\hat{y} = \hat{b}x + \hat{a}$ are calculated as follows:

$\hat{b} = \frac{\sum_{i=1}^{n}({x}_{i}-\overline{x})({y}_{i}-\overline{y})}{{\sum_{i=1}^{n}({x}_{i}-\overline{x})}^{2}} = \frac{\sum_{i=1}^{n}{x}_{i}{y}_{i}-n\overline{xy}}{\sum_{i=1}^{n}{x}_{i}^{2}-n{\overline{x}}^{2}}$

$\hat{a} = \overline{y} - \hat{b}\overline{x}$

$(2)$ Predict the number of students going out of the building during this break time for this class when the temperature is $-8^\circ\mathrm{C}$ (rounded to the nearest integer).

$(3)$ In order to encourage students to increase outdoor activity time, the school holds a tug-of-war competition with a 3-round, 2-win system (no draws). In the competition between Class A and Class B, the probability of Class A winning each round is $\frac{3}{5}$. Let the random variable $X$ represent the number of rounds Class A wins. Find the probability distribution and expectation of $X$.

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