Mathematics Question from JEE Main 2023 · 13 April, Shift 1

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