
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 factory packages a certain product in boxes, with each box containing 200 items. Before a box is delivered to customers, the products must be inspected. If any defective items are found, they are replaced with non-defective ones. During inspection, a random sample of 20 items is taken from the box for examination, and based on the results, a decision is made whether to inspect all the remaining items. Suppose the probability of each item being defective is $p (0 < p < 1)$, and the condition of each item being defective is independent of the others.
(1) Let $f(p)$ be the probability that exactly 2 out of the 20 items are defective. Find the value of $p$ at which $f(p)$ is maximized, denoted as $p_0$.
(2) After inspecting 20 items from a box, exactly 2 defective items are found. Using the value of $p_0$ determined in (1), assume $p = p_0$. Given that the inspection cost per item is 2 yuan, and if any defective item reaches a customer, the factory has to pay a compensation fee of 25 yuan per defective item.
(i) If the remaining items in the box are not inspected, the sum of the inspection fee and compensation fees for this box of products is denoted as $X$. Calculate $EX$.
(ii) Based on the expected value of the sum of the inspection fee and compensation fees, should all the remaining items in the box be inspected?

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