
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
Suppose the probabilities of four people, A, B, C, and D, needing to use a certain device on any given workday are 0.6, 0.5, 0.5, and 0.4, respectively. The events of each person needing the device are independent of each other.  
(1) Calculate the probability that at least three people need the device on the same workday.  
(2) The laboratory plans to purchase $k$ devices for A, B, C, and D to use. If the probability of "the number of people needing the device on the same workday is greater than $k$" is required to be less than 0.1, find the minimum value of $k$.

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