Problem:
Score : 400 points

Problem Statement

Mountaineers Mr. Takahashi and Mr. Aoki recently trekked across a certain famous mountain range.
The mountain range consists of N mountains, extending from west to east in a straight line as Mt. 1, Mt. 2, ..., Mt. N.
Mr. Takahashi traversed the range from the west and Mr. Aoki from the east.

The height of Mt. i is h_i, but they have forgotten the value of each h_i.
Instead, for each i (1 ≤ i ≤ N), they recorded the maximum height of the mountains climbed up to the time they reached the peak of Mt. i (including Mt. i).
Mr. Takahashi's record is T_i and Mr. Aoki's record is A_i.

We know that the height of each mountain h_i is a positive integer.
Compute the number of the possible sequences of the mountains' heights, modulo 10^9 + 7.

Note that the records may be incorrect and thus there may be no possible sequence of the mountains' heights.
In such a case, output 0.

Constraints

1 ≤ N ≤ 10^5

1 ≤ T_i ≤ 10^9

1 ≤ A_i ≤ 10^9

T_i ≤ T_{i+1} (1 ≤ i ≤ N - 1)

A_i ≥ A_{i+1} (1 ≤ i ≤ N - 1)

Input

The input is given from Standard Input in the following format:

N
T_1 T_2 ... T_N
A_1 A_2 ... A_N

Output

Print the number of possible sequences of the mountains' heights, modulo 10^9 + 7.

Sample Input 1

5
1 3 3 3 3
3 3 2 2 2

Sample Output 1

4

The possible sequences of the mountains' heights are:

1, 3, 2, 2, 2

1, 3, 2, 1, 2

1, 3, 1, 2, 2

1, 3, 1, 1, 2

for a total of four sequences.

Sample Input 2

5
1 1 1 2 2
3 2 1 1 1

Sample Output 2

0

The records are contradictory, since Mr. Takahashi recorded 2 as the highest peak after climbing all the mountains but Mr. Aoki recorded 3.

Sample Input 3

10
1 3776 3776 8848 8848 8848 8848 8848 8848 8848
8848 8848 8848 8848 8848 8848 8848 8848 3776 5

Sample Output 3

884111967

Don't forget to compute the number modulo 10^9 + 7.

Sample Input 4

1
17
17

Sample Output 4

1

Some mountain ranges consist of only one mountain.

Solution in Haskell (GHC 7.10.3) (judge verdict: Wrong Answer)

import Control.Monad
import Data.Maybe

main :: IO ()
main = do
  n <- readLn :: IO Int
  as <- getInts
  bs <- getInts
  let as'= decide as
      bs' = reverse $ decide (reverse bs)
      ms = zipWith (min) as bs
      xs = solve as' bs' ms
  print $ fromMaybe 0 (solve' <$> xs)
  

getInts :: IO [Int]
getInts = map read . words <$> getLine

decide :: [Int] -> [Int]
decide ns = go (zip ns (0 : init ns))
  where
    go [] = []
    go ((c, n):xs)
      | c == n = -1 : go xs
      | otherwise = c : go xs

solve :: [Int] -> [Int] -> [Int] -> Maybe [Int]
solve as bs mins = go as bs mins
  where
    go [] _ _ = Just []
    go (x:xs) (y:ys) (m:ms)
      | x == y && y == (-1) = fmap (m :) (go xs ys ms)
      | x == y = fmap (-1 :) (go xs ys ms)
      | x == -1 = fmap (-1 :) (go xs ys ms)
      | y == -1 = fmap (-1 :) (go xs ys ms)
      | otherwise = Nothing

solve' :: [Int] -> Int
solve' ns = go ns
  where
    go [] = 1
    go (n:ns)
      | n == -1 = go ns
      | otherwise = (n * go ns) `mod` 1000000007