HMMT November 2021 H10

Let $n$ be the answer to this problem. Suppose square $A B C D$ has side-length 3 . Then, congruent non-overlapping squares $E H G F$ and $I H J K$ of side-length $\frac{n}{6}$ are drawn such that $A, C$, and $H$ are collinear, $E$ lies on $B C$ and $I$ lies on $C D$. Given that $A J G$ is an equilateral triangle, then the area of $A J G$ is $a+b \sqrt{c}$, where $a, b, c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a+b+c$.