
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
On June 14, 2018, the 21st FIFA World Cup will kick off in Russia. A local sports channel organized fans to guess the outcomes of the matches for the top four popular teams: Germany, Spain, Argentina, and Brazil. Each fan can choose one team from the four, and currently, three people are participating in the guessing game.  
$(1)$ If each of the three people can choose any team and the selection of each team is equally likely, find the probability that exactly two teams are chosen by people.  
$(2)$ If one of the three people is a female fan, assuming the probability of the female fan choosing the German team is $\frac{1}{3}$ and the probability of a male fan choosing the German team is $\frac{2}{5}$, let $\xi$ be the number of people choosing the German team among the three. Find the probability distribution and the expected value of $\xi$.

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