A circle in the $xy$-plane is defined by the non-standard equation $x^2 + y^2 - 6kx + 8ky + 12k^2 - t = 0$, where $k$ and $t$ are positive constants. If the line $y = -4$ passes through the center of the circle, and the circumference of the circle is $12\pi\sqrt{3}$, what is the value of $t$ when $k = 1$?