Let $w = z\bar{z} + k_1 z + k_2 i\bar{z} + \lambda(1 + i)$, $k_1, k_2 \in \mathbb{R}$. If the curve $\text{Re}(w) = 0$ is the circle $C$ of radius $1$ in the first quadrant touching the line $y = 1$ and the $y$-axis, and the curve $\text{Im}(w) = 0$ intersects $C$ at $A$ and $B$, then $30(AB)^2$ is equal to: