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

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. $a<b$
3. $a+b=c$
4. $\mathrm{rad}(abc)<c$

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

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

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

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