
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
A basketball team plays 6 consecutive games against other teams, with each game resulting in a win or loss. Assume that the events of this team winning against other teams are independent, and the probability of winning is consistently $\frac{1}{3}$.
(1) Calculate the probability that the team loses two games before winning for the first time.
(2) Calculate the probability that the team wins exactly 3 out of the 6 games.
(3) Calculate the expected value (mean) of the number of games won by the team in these 6 games.

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