Score : 1300 points
You are playing a game and your goal is to maximize your expected gain. At the beginning of the game, a pawn is put, uniformly at random, at a position p\in\{1,2,\dots, N\}. The N positions are arranged on a circle (so that 1 is between N and 2).
The game consists of turns. At each turn you can either end the game, and get A_p dollars (where p is the current position of the pawn), or pay B_p dollar to keep playing. If you decide to keep playing, the pawn is randomly moved to one of the two adjacent positions p-1, p+1 (with the identifications 0 = N and N+1=1).
What is the expected gain of an optimal strategy?
Note: The "expected gain of an optimal strategy" shall be defined as the supremum of the expected gain among all strategies such that the game ends in a finite number of turns.
Input is given from Standard Input in the following format:
N A_1 A_2 \cdots A_N B_1 B_2 \cdots B_N
Print a single real number, the expected gain of an optimal strategy. Your answer will be considered correct if its relative or absolute error does not exceed 10^{-10}.
5 4 2 6 3 5 1 1 1 1 1
4.700000000000
4 100 0 100 0 0 100 0 100
50.000000000000
14 4839 5400 6231 5800 6001 5200 6350 7133 7986 8012 7537 7013 6477 5912 34 54 61 32 52 61 21 43 65 12 45 21 1 4
7047.142857142857
10 470606482521 533212137322 116718867454 746976621474 457112271419 815899162072 641324977314 88281100571 9231169966 455007126951 26 83 30 59 100 88 84 91 54 61
815899161079.400024414062