A polynomial function $P(x)$ has x-intercepts at $x=-4,x=0,$ and $x=3$. The graph of $P(x)$ touches the x-axis at $x=3$ and crosses the x-axis at $x=-4$ and $x=0$. As $x\to \infty$, $P(x)\to \infty$, and as $x\to -\infty$, $P(x)\to \infty$. If $P(x)$ can be written in the form $P(x)=C\cdot x^p(x-3{)}^m(x+4{)}^n$, where $C$ is a positive constant and $p,m,n$ are positive integers, what is the minimum possible value of $p+m+n$?