Cycles Must Be On Path

Not all positive cycles matter. Only cycles reachable from room $1$ that can reach room $n$ make the answer infinite. Run BFS or DFS from room $1$ to find all reachable rooms. Run BFS or DFS on the reverse graph from room $n$ to find all rooms that can reach room $n$.

A positive cycle matters only if it is in both sets. If Bellman-Ford finds a negative cycle, check if that cycle is in the intersection of these two sets.