Shuffle#2630
My office is quite far from my home, and every day I go there by bus. This takes a lot of time, and due to traffic jams, the duration of the journey to my office is not fixed. To make the best use of this time, I always carry my music player with me.
My music player has a shuffle mode. In this mode, it randomly selects a track that has not been played in that session and starts playing it. It will continue doing so until I stop playing, or all tracks in that session have been played. For the purpose of this problem, you can assume that all tracks are of equal duration. To select randomly, the player keeps a list of tracks that have already been played. That is, if I listen to one track now, in the next shuffle session that track will be marked as played. When selecting a track at random, the probability of selecting an old track (already listened) is twice as high as selecting a new (not yet listened) track.
I go to the office in the morning and return home at night. It takes at least minutes and at most minutes to reach the office. The travel time will always be an integer. You can assume that all possible travel times are equally likely (that is, if p = 2 and q = 4, and is the probability to reach the office in minutes, then P(2) = P(3) = P(4)). If all tracks have duration minute, how many days on average are required to listen to all the songs?
The first line contains an integer (1 \leq T \leq 10053), the number of test cases.
Each test case contains integers: (1 \leq N \leq 75), p_1, q_1, p_2, q_2 (0 \leq p_1,\, q_1,\, p_2,\, q_2 \leq 30,\ p_1 \leq q_1,\ p_2 \leq q_2), where is the number of songs, p_1 and q_1 are the minimum and maximum number of songs listened to during the day, and p_2 and q_2 are the minimum and maximum numbers listened to during the night.
For each test case, output the expected number of days required to listen to all songs. If it is impossible to listen to all songs, output IMPOSSIBLE. Errors less than 10^{-6} will be tolerated.
Example 1