Root of a random cubic in the unit interval

Let $M$ and $N$ be independent and uniform on $(0,1)$, and fix $0<c<2$. Write $p(c)$ for the chance that the equation $Mx^3+Nx=c$ is satisfied by some $x$ strictly between $0$ and $1$. It is possible to express $p(c)=1-\Big[(ac^2+bc+d)\,\mathbf{1}_{\{0<c<1\}}+(rc^2+sc+t)\,\mathbf{1}_{\{1<c<2\}}\Big]$ for rational constants $a,b,d,r,s,t$. What is $a+b+d+r+s+t$?

$1$. The map $x\mapsto Mx^3+Nx$ is strictly increasing from $0$ at $x=0$ to $M+N$ at $x=1$, so a root exists iff $M+N>c$. Hence the bracket is the CDF of $M+N$: $\tfrac{c^2}{2}$ on $(0,1)$ and $-\tfrac{c^2}{2}+2c-1$ on $(1,2)$. The coefficients sum to $\tfrac12+0+0-\tfrac12+2-1=1$.