In triangle $DEF,$ $\angle D = 30^\circ,$ $\angle F = 60^\circ,$ and $EF = 6.$ Find $DE + DF.$ [asy] unitsize(2 cm); pair D, E, F; D = (0,sqrt(3)); E = (0,0); F = (1,0); draw(D--E--F--cycle); label("$D$", D, N); label("$E$", E, SW); label("$F$", F, SE); label("$6$", (E + F)/2, S); [/asy]
