
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
Consider keys numbered 1, 2, ..., 10 and boxes numbered 1, 2, ..., 10, where each key can only open the box with the same number. All keys are placed into these boxes, with one key in each box, and then the boxes are locked. What is the minimum number of different ways to place the keys such that at least two boxes must be opened to unlock all the boxes? Also, how many ways are there to arrange the keys such that exactly two boxes must be opened to unlock all the boxes?

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