The radical of $n$, $\operatorname{rad}(n)$, is the product of distinct prime factors of $n$. For example, $504 = 2^3 \times 3^2 \times 7$, so $\operatorname{rad}(504) = 2 \times 3 \times 7 = 42$.

We shall define the triplet of positive integers $(a, b, c)$ to be an abc-hit if:

1. $\gcd(a, b) = \gcd(a, c) = \gcd(b, c) = 1$
2. $a \lt b$
3. $a + b = c$
4. $\operatorname{rad}(abc) \lt c$

For example, $(5, 27, 32)$ is an abc-hit, because:

1. $\gcd(5, 27) = \gcd(5, 32) = \gcd(27, 32) = 1$
2. $5 \lt 27$
3. $5 + 27 = 32$
4. $\operatorname{rad}(4320) = 30 \lt 32$

It turns out that abc-hits are quite rare and there are only thirty-one abc-hits for $c \lt 1000$, with $\sum c = 12523$.

Find $\sum c$ for $c \lt 120000$.