
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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In a certain city, five joint math competitions are held for high school sophomores, and if a student ranks in the top 20 in the city twice out of these competitions, they will qualify for provincial training and do not need to participate in the remaining competitions, though each student can participate in a maximum of five competitions. It is stipulated that if the student's performance in the first four competitions does not rank in the top 20 in the city, they cannot participate in the fifth competition. Assume that the probability of a student ranking in the top 20 in each competition is $\frac{1}{4}$, and the performances in each competition are independent of each other.

(1) Find the probability that the student qualifies for provincial training.
(2) If the student qualifies for provincial training or completes all five competitions, let $\xi$ denote the number of competitions the student participates in. Determine the distribution of $\xi$ and find its expected value.

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