
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
(This question is worth 12 points) Six balls, numbered 1 through 6, of the same size are to be placed into six boxes, also numbered 1 through 6, with each box holding exactly one ball.  
(I) If ball number 1 can only be placed in box number 1, and ball number 2 can only be placed in box number 2, how many different arrangements are there?  
(II) If ball number 3 can only be placed in box number 1 or 2, and ball number 4 cannot be placed in box number 4, how many different arrangements are there?  
(III) If balls number 5 and 6 can only be placed into two boxes with consecutive numbers, how many different arrangements are there?

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