
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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The English college entrance examination consists of two parts: listening and speaking test with a full score of $50$ points, and the English written test with a full score of $100$ points. The English listening and speaking test is conducted twice. If a student takes both tests, the higher score of the two tests will be taken as the final score. If the student scores full marks in the first test, they will not take the second test. To prepare for the English listening and speaking test, Li Ming takes English listening and speaking mock exams every week. The table below shows his scores in $20$ English listening and speaking mock exams before the first actual test.

Assumptions: 
1. The difficulty level of the mock exams is equivalent to that of the actual exam.
2. The difficulty level of the two actual listening and speaking tests is the same.
3. If Li Ming continues to practice listening and speaking after not scoring full marks in the first test, the probability of scoring full marks in the second test can reach $\frac{1}{2}$.

| $46$ | $50$ | $47$ | $48$ | $49$ | $50$ | $50$ | $47$ | $48$ | $47$ |
|------|------|------|------|------|------|------|------|------|------|
| $48$ | $49$ | $50$ | $49$ | $50$ | $50$ | $48$ | $50$ | $49$ | $50$ |

$(Ⅰ)$ Let event $A$ be "Li Ming scores full marks in the first English listening and speaking test." Estimate the probability of event $A$ using frequency.

$(Ⅱ)$ Based on the assumptions in the problem, estimate the maximum probability of Li Ming scoring full marks in the English college entrance examination listening and speaking test.

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