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]

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