Segments from the Midpoint of an Equilateral Triangle

In the equilateral triangle $ABC$ with side length $2$ shown in the figure, $D$ is the midpoint of $BC$, and $E$, $F$ are moving points on sides $AB$ and $AC$ respectively.

1. If $\angle EDF=120^{\circ}$, prove that $AE+AF$ is a constant.

2. If $\angle EDF=60^{\circ}$, is $AE+AF$ still a constant? If so, give a proof; if not, find the range of values of $AE+AF$.