
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
There are two research groups, $A$ and $B$, in the laboratory. According to the statistical data, the probabilities of the two groups $A$ and $B$ completing their research tasks are $P_1= \frac{2}{3}$ and $P_2$, respectively. It is assumed that each research project contains two tasks to be completed, and the completion of each task is independent of the other. If, in one research project, both groups complete an equal number of tasks and each completes at least one task, then the research is termed as an "Advanced Harmonious Laboratory."

(I) If $P_2= \frac{1}{2}$, find the probability that the laboratory is awarded the title of "Advanced Harmonious Laboratory" upon the completion of one research project;
(II) Suppose that the number of times the laboratory is awarded the title of "Advanced Harmonious Laboratory" after completing six research projects is $\xi$. Find the range of values for $P_2$ for which $E\xi \geq 2.5$.

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