
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 $4$ balls numbered $1$, $2$, $3$, $4$ into $4$ boxes numbered $1$, $2$, $3$, $4$.<br/>$(1)$ How many ways are there to place the balls?<br/>$(2)$ Each box can have at most one ball, how many ways are there to place the balls?<br/>$(3)$ If exactly one box is empty, how many ways are there to place the balls?<br/>$(4)$ If the $4$ different balls are replaced with $4$ identical balls and exactly one box is empty, how many ways are there to place the balls?

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