
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 technology company encountered a technical challenge and urgently formed two teams, Team A and Team B, to work on the problem separately for a month. It was decided that the team successfully solving the technical problem at the end of the period would receive a reward. It is known that the probability of Team A solving the problem by the deadline is $\frac{2}{3}$, and for Team B, the probability is $\frac{3}{4}$.
(1) Let $\xi$ be the number of teams rewarded at the end of the challenge period. Find the probability distribution of $\xi$ and $E\xi$ (the expected value of $\xi$).
(2) Let $\eta$ be the square of the difference between the number of rewarded teams and the number of unrewarded teams at the end of the challenge period. Let event C be defined as "the function $f(x) = |\eta - \frac{7}{2}|^x$ decreases monotonically in its domain." Calculate the probability of event C.

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