
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 blogger on a short video platform mainly showcases rural life in their videos, such as going to the market, fishing, and raising chickens, attracting many viewers. The blogger conducted live broadcasts to promote a certain agricultural product from their hometown, and obtained the following data on sales $y$ (in million boxes) and unit price $x$ (in yuan per box) through 5 trial sales:

| $x$   | $5$  | $5.5$ | $6$  | $6.5$ | $7$  |
|-------|------|-------|------|-------|------|
| $y$   | $50$ | $48$  | $43$ | $38$  | $36$ |

$(1)$ Based on the data above, find the regression line equation for $y$ in terms of $x$;  
$(2)$ Among all customers, a survey was conducted by randomly selecting some customers (a large number of people). Half of them rated their experience as "very good," while "good" and "unsatisfactory" experiences accounted for $30\%$ and $20\%$ respectively. Then, 5 lucky customers were randomly selected from all customers to receive gifts. Let the random variable $X$ represent the number of "very good" experiences among the 5 selected customers. Find the probability distribution and variance of $X$.

Given formula: The empirical regression equation is $\hat{y}=\hat{b}x+\hat{a}$, where $\hat{b}=\frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})({{y_i}-\overline{y}})}}}{{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}}}}=\frac{{\sum_{i=1}^n{{x_i}{y_i}-n\overline{x}\overline{y}}}}{{\sum_{i=1}^n{x_i^2-n{{\overline{x}}^2}}}},\hat{a}=\overline{y}-\hat{b}\overline{x}$.

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