
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 a knowledge competition held by a school, the first round consists of four problems, A, B, C, and D. The rules are as follows:
1. Each contestant starts with 10 points. Correct answers to problems A, B, C, and D add 1, 2, 3, and 6 points respectively, while a wrong answer to any problem deducts 2 points.
2. After each problem, the scoreboard shows the cumulative score. If the cumulative score falls below 8 points, the contestant is eliminated; if it is 14 points or more, the contestant advances to the next round. If the cumulative score is less than 14 points after answering all four problems, the contestant is also eliminated.
3. Each contestant answers the problems in the order A, B, C, and D until they are finished or eliminated.
Suppose contestant A has the probabilities of answering problems A, B, C, and D correctly as $\frac {3}{4}, \frac {1}{2}, \frac {1}{3}, \frac {1}{4}$, respectively, and each answer is independent of the others.
(i) Calculate the probability that contestant A advances to the next round.
(ii) Let $\xi$ represent the number of problems contestant A answers in this round. Find the probability distribution and the expected value, $E\xi$.

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