
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

...

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system
Currently, a new round of technological revolution and industrial change is flourishing, with the rapid development of new generation information technologies represented by blockchain. Data on the total number of blockchain companies in a certain location over the past 6 years are collected as shown in the table below:

| Year | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
|------|------|------|------|------|------|------|
| Number $x$ | 1 | 2 | 3 | 4 | 5 | 6 |
| Total number of companies $y$ (in hundreds) | 50 | 78 | 124 | 121 | 137 | 352 |

$(1)$ If the model $y=ae^{bx}$ is used to fit the relationship between $y$ and $x$, based on the provided data, determine the empirical regression equation for $y$ and $x$;<br/>$(2)$ In order to promote cooperation and development among companies, the blockchain joint headquarters decides to hold an information technology competition, inviting three blockchain companies, A, B, and C, to participate. The rules of the competition are as follows: ① Each match has two companies participating, and a winner is determined; ② The winning company of each match competes with the company that did not participate in that match in the next match; ③ In the competition, if one company wins two matches first, the competition ends, and that company becomes the "champion" of this information technology competition. It is known that in each match, the probability of A winning against B is $\frac{1}{2}$, the probability of A winning against C is $\frac{1}{3}$, and the probability of B winning against C is $\frac{3}{5}$. If the first match is between A and B, find the probability of company A becoming the "champion".

Reference data: $\sum_{i=1}^6{{u_i}}=28.5, \sum_{i=1}^6{{x_i}{u_i}}=106.05$, where $u_{i}=\ln y_{i}$<br/>Reference formula: For a set of data $(x_{i}, y_{i}) (i=1,2,3,\ldots,n)$, the least squares estimates of the slope and intercept of the empirical regression line $\hat{y}=\hat{b}x+\hat{a}$ are $\hat{b}=\frac{{\sum_{i=1}^n{x_i}{y_i}-n\overline{x}\cdot\overline{y}}}{{\sum_{i=1}^n{x_i^2}-n\overline{x}^2}}, \hat{a}=\overline{y}-\hat{b}\overline{x}$

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