Trigonometry · Superchallenge problem

Unit vectors $\mathbf e_1$ and $\mathbf e_2$ make an angle of $60^\circ$. Real numbers $x$ and $y$ are chosen so that the vector $x\,\mathbf e_1 + y\,\mathbf e_2$ has length $1$. Find the largest possible value of $x + y$.