So I've been trying to solve this problem recently.
My current solution is this:
For each contiguous dark segment, extend a line that passes through the dark segment's endpoints. The position to place the lantern must lie on all of these lines.
Second, for the walls that are completely illuminated, we can create half planes from that, with which we can find points that can reach all walls. This process will give us a polygon. The position obtained from using the dark segments must lie inside this polygon.
However, I am not satisfied that this is the right solution. Compared to the other problems in this contest, it seems far more difficult to implement. In addition, there seems to be many corner cases involved in this problem as well, leading to an inelegant solution. I fear I have missed an obvious solution, and decided to post here to make sure.