Length of a Perpendicular Segment in a Triangle

An equilateral triangle $ \triangle ABC $ has side length 2. Point $ D $ lies on segment $ \overline{AC} $ such that $ \angle DBC = 45^\circ $. The foot of the perpendicular from $ D $ to $ \overline{BC} $ is point $ E $. What is the length of $ \overline{DE} $? The answer is in the form $a-b\sqrt{c}$ where $a$, $b$, $c$ are positive integers, and $c$ is square-free. Input the value of $a+b+c$.