
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 certain organic fruit plantation is experimenting with the cultivation of a particular fruit that must be packaged in boxes before sale, with each box containing 80 fruits. Before delivering the box to customers, the fruits must be inspected according to a set standard. If any unqualified fruits are found, they will be replaced with qualified ones. During the inspection, 10 fruits are randomly selected from the box for inspection, and a decision is made whether to inspect the remaining fruits based on the inspection results. Let the probability of each fruit being unqualified be p (0 < p < 1), and whether each fruit is unqualified is independent of each other.

(I) Denote the probability of having exactly 2 unqualified fruits in the 10 selected fruits as f(p), find the value of p (denoted as p0) when f(p) reaches its maximum.

(II) Now, after inspecting 10 fruits from a box, it was found that exactly 2 were unqualified. Using the p0 determined in (I) as the value of p. Given that the inspection cost for each fruit is 1.5 yuan, if an unqualified fruit ends up in the customer's hands, the plantation must pay a compensation fee of a yuan for each unqualified fruit (a ∈ N*).

(i) If the remaining fruits in the box are not inspected, denote the sum of the inspection fee and compensation fee for this box as X, find E(X).

(ii) Using the expected value of the sum of inspection and compensation fees as the decision criterion, what is the minimum compensation fee per unqualified fruit that would prompt the plantation to inspect all the remaining fruits in the box?

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