Let $P(x)$ be the unique polynomial of minimal degree with the following properties:

• $P(x)$ has a leading coefficient $1$,
• $1$ is a root of $P(x)-5$,
• $3$ is a root of $P(x-1)$,
• $2$ is a root of $P(2x)$, and
• $6$ is a root of $6P(x)$.

The roots of $P(x)$ are integers, with one exception. The root that is not an integer can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. What is $m+n$?