
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
In 2019, a city's math olympiad trial reform was as follows: The city holds 5 joint competitions in the second year of high school. If a student achieves a top-20 ranking in the city in 2 out of those 5 competitions, they can join the provincial team training and do not need to participate in the remaining competitions. Each student can participate in up to 5 competitions, with the condition that if a student does not achieve a top-20 ranking in the first 4 competitions, they cannot participate in the 5th competition. Assume the probability of the student achieving a top-20 ranking in the city in each competition is $\frac{1}{4}$, and the results of each competition are independent.

(1) Find the probability that the student joins the provincial team.
(2) If the student joins the provincial team or completes 5 competitions, denote the number of competitions the student participates in as $\xi$. Find the distribution and the expected value of $\xi$.

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