
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 bank queue problem. If one person spends one minute waiting in line, we'll say that one person-minute is wasted. There are eight people in the queue, five of whom are planning simple operations that take 1 minute each, while the remaining three plan long operations that take 5 minutes. Find:

a) the minimum and maximum possible total number of wasted person-minutes;

b) the expected value of the number of wasted person-minutes, assuming the customers join the queue in a random order.

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