
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
Four students went to the library together after class and put their backpacks together. Later, the power went out, and everyone randomly picked up a backpack and left the library. Calculate the probabilities of the following events:
- $(1)$ Exactly two students picked up the correct backpacks.
- $(2)$ At least two students picked up the correct backpacks.
- $(3)$ All the backpacks were picked up incorrectly.

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