
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 certain competition that proceeds in three stages—preliminary, semifinal, and final—each contestant must answer a question at each stage. It is required that those who answer correctly proceed to the next stage, while the others are eliminated. It is known that the probabilities of a contestant passing the preliminary, semifinal, and final are $ \frac {2}{3}, \frac {1}{2},$ and $ \frac {1}{3}$ respectively, and the events of passing each stage are independent of each other.
$(1)$ Calculate the probability that the contestant is eliminated during the semifinal stage;
$(2)$ Let the random variable $\xi$ represent the number of questions the contestant answers in the competition. Find the probability distribution and expected value of $\xi$.

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