
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 shooting team consists of two shooters, A and B. Shooter A has a hit rate of $P_{1}= \frac{2}{3}$, while Shooter B has a hit rate of $P_{2}$. In a marksmanship competition, each shooter fires two shots to complete one test. If, in a single test, both shooters hit the target an equal number of times and each hits at least once, the team is dubbed an "Advanced Harmonious Group."
(1) If $P_{2}= \frac{1}{2}$, what is the probability that the team is recognized as an "Advanced Harmonious Group" in a single test?
(2) Suppose a test is conducted once a month in the year 2011, and let ξ denote the number of times the team is recognized as an "Advanced Harmonious Group" out of the 12 tests. If the expected value Eξ≥5, determine the range of values for $P_{2}$.

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