
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 introduced a reform in its high school math competition: Students in the second year of high school compete in five joint competitions. If a student ranks in the top 20 in the city in any two of these competitions, they qualify for provincial team training and do not need to participate in the remaining competitions. A student can participate in a maximum of five competitions. It is specified that if a student does not rank in the top 20 in the city in any of the first four competitions, they cannot participate in the fifth competition. Suppose the probability that a student ranks in the top 20 in the city in any given competition is $\frac{1}{4}$, and the outcomes of each competition are independent.

(1) Find the probability that the student qualifies for the provincial team.

(2) If the student either qualifies for the provincial team or finishes all five 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}
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