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 $P(x)-1$,  
$2$ is a root of $P(x-2)$,  
$3$ is a root of $P(3x)$, and  
$4$ is a root of $4P(x)$.  
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. Find $m+n$.

Let's think step by step and output the final answer within \boxed{}.
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