Line $l$ has equation $x-3y+24=0$. This line is rotated $45^\circ$ counterclockwise about the point $(12,12)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k$?

1. If $\tan\alpha$ is the slope of $l$ ($\tan\alpha=\frac13$), the new slope is $\tan(45^\circ+\alpha)=\dfrac{1+\frac13}{1-\frac13}=2$.
2. Line $k$: $y-12=2(x-12)$.
3. Setting $y=0$: $x=\boxed6$.