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.