
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
Place 5 balls numbered 1, 2, 3, 4, 5 into 5 boxes also numbered 1, 2, 3, 4, 5.  
(1) How many ways are there to do this?  
(2) If each box can hold at most one ball, how many ways are there?  
(3) If exactly one box is to remain empty, how many ways are there?  
(4) If each box contains one ball, and exactly one ball's number matches its box's number, how many ways are there?  
(5) If each box contains one ball, and at least two balls' numbers match their boxes' numbers, how many ways are there?  
(6) If the 5 distinct balls are replaced with 5 identical balls, and exactly one box is to remain empty, how many ways are there?  
(Note: For all parts, list the formula before calculating the value, otherwise, points will be deducted.)

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