
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
Pure electric vehicles, hybrid electric vehicles, and fuel cell electric vehicles are all new energy vehicles. In recent years, the number of new energy vehicles in a certain region has been growing rapidly. The table below shows the number of new energy vehicles and pure electric vehicles in the region from 2018 to 2022 (unit: 10,000 vehicles). The corresponding years from 2018 to 2022 are numbered as 1 to 5:

| Year Number $x$ | 1   | 2   | 3   | 4   | 5   |
|-----------------|-----|-----|-----|-----|-----|
| New Energy Vehicles $y$ | 1.5 | 2.6 | 3.4 | 4.9 | 7.8 |
| Pure Electric Vehicles $z$ | 1.3 | 2.1 | 2.8 | 4.0 | 6.4 |

$(1)$ Based on the data in the table, the exponential function model $y=a\cdot b^{x}$ can be used to fit the relationship between $y$ and $x$. Please establish the regression equation for $y$ with respect to $x$ $(a$ and $b$ accurate to $0.1)$, and predict whether the number of new energy vehicles in the region will exceed 10,000 vehicles in 2023.

$(2)$ From the data in the table, it can be seen that from 2018 to 2022, the proportion of pure electric vehicles in the number of new energy vehicles in the region exceeded 80%, indicating that pure electric vehicles have always been the mainstream product of new energy vehicles. If individuals A, B, and C each randomly select one year from 2018 to 2022 (repeated selection allowed), and the number of years satisfying $y-z \gt 0.8$ is denoted as $X$, find the distribution and mathematical expectation of $X$.

Reference data:

| $\overline{v}$ | $\sum_{i=1}^{5}{{x}_{i}}{v}_{i}$ | $e^{0.089}$ | $e^{0.387}$ | $1.5^{6}$ |
|----------------|---------------------------------|-------------|-------------|-----------|
| 1.25           | 22.62                           | 1.1         | 1.5         | 11.4      |

Here, $v_{i}=\ln y_{i}$, $\overline{v}=\frac{1}{5}\sum_{i=1}^{5}{{v}_{i}}$.

Reference formulas: For a set of data $((u_{1}, v_{1}), (u_{2}, v_{2}), \ldots, (u_{n}, v_{n}))$, the least squares estimation formulas for the slope and intercept of the regression line $\hat{v}=\hat{\alpha}+\hat{\beta}u$ are respectively $\hat{\beta}=\frac{\sum_{i=1}^{n}{{u}_{i}}{v}_{i}-n\overline{u}\overline{v}}{\sum_{i=1}^{n}{{u}_{i}^{2}}-n{\overline{u}}^{2}}$ and $\hat{a}=\overline{v}-\hat{\beta}\overline{u}$.

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