The solution to this problem is a modified version of dijkstra. It is almost the same, just when you calculate the time to travel a path, you must also add waiting for the gate at the end.

I took a dijkstra solution from geekforgeek online: https://www.geeksforgeeks.org/python-program-for-dijkstras-shortest-path-algorithm-greedy-algo-7/

I modified their code to solve this problem, and then I used pwn tools to interface with the challenge and solve all the things. Solve 50 levels, and it gives the flag. It finishes with plenty of time to spare.

Original writeup (https://github.com/xMestas/ctfWriteups/tree/master/fireshell2020/dungeon_escape).