
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
From the digits $1$ to $9$, select $3$ even numbers and $4$ odd numbers.

$(1)$ How many different seven-digit numbers can be formed with these selected numbers without repetition?

$(2)$ Among these seven-digit numbers, how many have the $3$ even numbers arranged together?

$(3)$ Among the seven-digit numbers in $(1)$, how many have both the even numbers arranged together and the odd numbers arranged together?

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