
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
The environmental protection department conducts pollution inspections on 5 paper mills. If the inspection fails, rectification is required. If the rectification fails upon re-inspection, the mill will be shut down. Assume that the inspection results of each paper mill are independent of each other, and the probability of passing the inspection before rectification is $\frac{1}{2}$, and the probability of passing the inspection after rectification is $\frac{4}{5}$. Find:  
(I) The probability that exactly two paper mills must be rectified;  
(II) The probability that at least one paper mill will be shut down;  
(III) The average number of paper mills that need rectification (where $(\frac{9}{10})^5 \approx \frac{59}{100}$).

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