
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
A certain TV station held a TV Olympic knowledge contest, which was divided into preliminary and final rounds. In order to increase the interest of the program, the preliminary round was conducted in a way that each contestant could choose a question to answer, with a maximum of 5 chances. A contestant would stop the preliminary round if they answered 3 questions correctly or incorrectly. Those who answered 3 questions correctly would directly enter the final round, while those who answered 3 questions incorrectly would be eliminated. It is known that the correct rate of contestant A is $\frac{2}{3}$.  
$(Ⅰ)$ Find the probability that contestant A can enter the final round;  
$(Ⅱ)$ Let the number of questions answered by contestant A in the preliminary round be $\xi$, write down the distribution of $\xi$, and find the mathematical expectation of $\xi$.

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