HMMT 2018 G4

A paper equilateral triangle of side length 2 on a table has vertices labeled $A, B, C$. Let $M$ be the point on the sheet of paper halfway between $A$ and $C$. Over time, point $M$ is lifted upwards, folding the triangle along segment $B M$, while $A, B$, and $C$ remain on the table. This continues until $A$ and $C$ touch. Find the maximum volume of tetrahedron $A B C M$ at any time during this process.