
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
Every spring, the sea of rapeseed flowers in Wuyuan attracts tens of thousands of tourists, making rapeseed flowers a characteristic landscape of "China's most beautiful countryside." In March, the rapeseed flower sea in Huangling, Wuyuan enters its best viewing period. The data on the number of tourists visiting Huangling for nearly seven years $(2015$ represented by $x=1$, $2016$ represented by $x=2)$ is as shown in the table below:

| $x$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|-----|---|---|---|---|---|---|---|
| Number of Tourists $y$ (in ten thousand) | 29 | 33 | 36 | 44 | 48 | 52 | 59 |

$(1)$ If $y$ has a strong linear relationship with $x$, find the linear regression equation of $y$ with respect to $x$, $y=\hat{b}x+\hat{a}$, and predict the number of tourists in Huangling in 2022;<br/>$(2)$ To maintain order in tourism, four civil servants $A$, $B$, $C$, $D$ need to be on duty at various scenic spots. It is known that the probabilities of $A$, $B$, $C$ going on duty in Huangling are all $\frac{2}{3}$, and the probability of $D$ going on duty in Huangling is $\frac{1}{3}$. The decision of each civil servant to go on duty in Huangling is independent. Let $X$ represent the number of people among these four who are on duty in Huangling. Find the distribution table and mathematical expectation of $X$.<br/>Reference formulas: $\hat{b}=\frac{{\sum_{i=1}^n{({{x_i}-\overline{x}})}({{y_i}-\overline{y}})}}{{\sum_{i=1}^n{{{({{x_i}-\overline{x}})}^2}}}}$, $\hat{a}=\overline{y}-\hat{b}\overline{x}$. Reference data: $\sum_{i=1}^7{{y_i}}=301$, $\sum_{i=1}^7{({{x_i}-\overline{x}})}({{y_i}-\overline{y}})=140$.

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