
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
Create a five-digit number using the digits $0$, $1$, $2$, $3$, $4$ without repetition.
$(I)$ How many odd numbers can be formed?
$(II)$ How many positive integers can be formed such that $1$ and $3$ are not adjacent?
$(III)$ How many positive integers can be formed such that $1$ is not in the ten-thousands place and $2$ is not in the units place?

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