<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 Given any two positive integers, a certain operation (denoted by the operator $\oplus$) is defined 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, $m \oplus n = m \cdot n$. The number of elements in the set $M = {(a, b) \mid a \oplus b = 12, a, b \in \mathbb{N}^*}$ is $\_\_\_\_\_\_$. Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
15
Given any two positive integers, a certain operation (denoted by the operator $\oplus$) is defined 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, $m \oplus n = m \cdot n$. The number of elements in the set $M = {(a, b) \mid a \oplus b = 12, a, b \in \mathbb{N}^*}$ is $\_\_\_\_\_\_$.
test