
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
For any two positive integers, define the operation $\oplus$ as follows: when $m$ and $n$ are both positive even numbers or both positive odd numbers, $m \oplus n = m + n$; when one of $m$ and $n$ is a positive even number and the other is a positive odd number, $m \oplus n = m \times n$. For example, $4 \oplus 6 = 4 + 6 = 10$, $3 \oplus 7 = 3 + 7 = 10$, $3 \oplus 4 = 3 \times 4 = 12$. Under this definition, the set $M = \{(a, b) \mid a \oplus b = 12, a, b \in \mathbb{N}^+\}$ has ___ elements.

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