A Circumcenter Distance Minimum

Three lines $l_1,l_2,l_3$ are pairwise parallel; the distance between $l_1$ and $l_2$ is $1$, between $l_2$ and $l_3$ is $\dfrac 12$, and between $l_1$ and $l_3$ is $\dfrac 32$ (so $l_2$ lies between $l_1$ and $l_3$). Points $A,B$ are fixed on $l_1$ with $AB=2$, and $M,N$ are moving points on $l_2$ with $MN=2$. Let $C$ be the circumcenter of $\triangle AMN$, and let $d$ be the distance from $C$ to $l_3$. Find the minimum value of $d+|BC|$.