Find all 4-tuples (a, b, c, d) of distinct positive integers so that a < b < c < d and 1/a + 1/b + 1/c + 1/d = 1.

My attempt:

If a = 1, 1/b + 1/c + 1/d = 0, which is impossible.

If a >= 3, b >= 4, c >= 5, d >= 6 implies that 1/a + 1/b + 1/c + 1/d < 1/3 + 1/4 + 1/5 + 1/6 < (20 + 15 + 12 + 10)/60 = 57/60 < 1.

So a = 2, 1/b + 1/c + 1/d = 1/2 and b >= 3.

How do I proceed?