tl;dr Make requests for http://hack.bckdr.in/INFINITE-PATHS/path/..../path//path
Each request to http://hack.bckdr.in/INFINITE-PATHS/path/path/path/path/path/path/... returns Keep roaming buddy!
Surprisingly http://hack.bckdr.in/INFINITE-PATHS//path gives us A part of the flag(length=50) is: "Y"
It turns out that we just have to make 50 requests to "http://hack.bckdr.in/INFINITE-PATHS"+"/path"*N +"//path" to get the full flag