Question 45
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 $BM$, while $A$, $B$, and $C$ remain on the table. This continues until $A$ and $C$ touch. Find the maximum volume of tetrahedron $ABCM$ at any time during this process.