
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
(The full score of this question is 14 points)  
Suppose there are five balls numbered 1, 2, 3, 4, 5, and five boxes also numbered 1, 2, 3, 4, 5. Now, these five balls are to be placed into the five boxes.  
(1) If only one box is left empty, how many different ways can the balls be placed?  
(2) If no box is left empty, but the numbers on the balls do not all match the numbers on the boxes, how many different ways can the balls be placed?  
(3) If one ball is placed in each box, and at least two balls have the same number as their boxes, how many different ways can the balls be placed?

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