output_description: 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}.
  
  * * *
input_description: 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
attempt: n = int(input())
  print(n**2)
  
problem_description: Statement
  
  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.
samples: [{"input": "5\n    4 2 6 3 5\n    1 1 1 1 1", "output": "4.700000000000\n    \n\n* * *"}, {"input": "4\n    100 0 100 0\n    0 100 0 100", "output": "50.000000000000\n    \n\n* * *"}, {"input": "14\n    4839 5400 6231 5800 6001 5200 6350 7133 7986 8012 7537 7013 6477 5912\n    34 54 61 32 52 61 21 43 65 12 45 21 1 4", "output": "7047.142857142857\n    \n\n* * *"}, {"input": "10\n    470606482521 533212137322 116718867454 746976621474 457112271419 815899162072 641324977314 88281100571 9231169966 455007126951\n    26 83 30 59 100 88 84 91 54 61", "output": "815899161079.400024414062"}]