
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
Using the digits 1, 2, 3, and 4 (with digits allowed to be repeated), how many $n$-digit numbers ($n \geqslant 2$) can be formed that contain an odd number of 1's, an even number of 2's, and at least one 3?

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