
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]


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A certain seedling, from the day of observation, has a height of $ycm$ on the $x$-th day, and some data are recorded in the table below:

| Day $x$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---------|---|---|---|---|---|---|---|
| Height $ycm$ | 0 | 4 | 7 | 9 | 11 | 12 | 13 |

When a scatter plot of this data is made, it is found that the relationship between $y(cm)$ and $x$(days) approximately satisfies the equation $y=bx+a$.

$(1)$ Using a simple linear regression model, based on the given data, find the linear regression equation of $y$ with respect to $x$, $\hat{y}=\hat{b}x+\hat{a}$.

$(2)$ In the scatter plot of this data, student A randomly circled three points, and let $\xi$ be the number of points among these three points whose seedling height is greater than the average height $\overline{y}$, where $\overline{y}$ is the average height of the seedlings given in the table. Find the probability distribution and the expected value of the random variable $\xi$.

Given: The least squares estimation formulas for the slope and intercept in the regression equation $y=\hat{b}x+\hat{a}$ are $\hat{b}=\frac{{\sum_{i=1}^n{{x_i}{y_i}-n\overline{x}\overline{y}}}{{\sum_{i=1}^n{x_i^2}-n{{\overline{x}}^2}}}$ and $\hat{a}=\overline{y}-b\overline{x}$.

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