
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

Let \( A_{1}, A_{2}, \ldots, A_{i}, \ldots, A_{n} \) be n mutually independent events. The probability of occurrence of each event \( A_{i} \) is \( p_{i} \) for \( i=1,2, \ldots, n \). In a trial, any combination of these events can occur, or none of them can occur. Find the probability of the occurrence of at least one of the events \( A_{i} \) given the probabilities \( p_{i} \):
a) are different;
b) are the same.

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