HMMT November 2024 U31

Positive integers $a, b$, and $c$ have the property that $\operatorname{lcm}(a, b), \operatorname{lcm}(b, c)$, and $\operatorname{lcm}(c, a)$ end in 4, 6, and 7, respectively, when written in base 10. Compute the minimum possible value of $a+b+c$.