problem link: 26D. Tickets
editorial: problem D

I have read the editorial of the problem but I'm not getting the reflection principle part. Especially how the reflection principle works here and why every graph ending at the point (m + n,  - 2 - n - k + m) will intersect with the line y = -1?

If anyone here knows the reason please explain me or if you have solved it with another approach can share it with me.
Thank you.