Proof Writing & Problem Solving · Superchallenge problem

Positive integers $a$, $b$, $c$ satisfy $\dfrac1a + \dfrac1b + \dfrac1c < 1$. A careful write-up organises the search with "without loss of generality $a \le b \le c$" and then splits into cases on $a$ and $b$. What is the largest possible value of $\dfrac1a + \dfrac1b + \dfrac1c$?