Problem:
Score : 800 points

Problem Statement

Takahashi Lake has a perimeter of L. On the circumference of the lake, there is a residence of the lake's owner, Takahashi.
Each point on the circumference of the lake has a coordinate between 0 and L (including 0 but not L), which is the distance from the Takahashi's residence, measured counter-clockwise.

There are N trees around the lake; the coordinate of the i-th tree is X_i. There is no tree at coordinate 0, the location of Takahashi's residence.

Starting at his residence, Takahashi will repeat the following action:

If all trees are burnt, terminate the process.

Specify a direction: clockwise or counter-clockwise.

Walk around the lake in the specified direction, until the coordinate of a tree that is not yet burnt is reached for the first time.

When the coordinate with the tree is reached, burn that tree, stay at the position and go back to the first step.

Find the longest possible total distance Takahashi walks during the process.

Partial Score

A partial score can be obtained in this problem:

300 points will be awarded for passing the input satisfying N \leq 2000.

Constraints

2 \leq L \leq 10^9

1 \leq N \leq 2\times 10^5

1 \leq X_1 < ... < X_N \leq L-1

All values in input are integers.

Input

Input is given from Standard Input in the following format:

L N
X_1
:
X_N

Output

Print the longest possible total distance Takahashi walks during the process.

Sample Input 1

10 3
2
7
9

Sample Output 1

15

Takahashi walks the distance of 15 if the process goes as follows:

Walk a distance of 2 counter-clockwise, burn the tree at the coordinate 2 and stay there.

Walk a distance of 5 counter-clockwise, burn the tree at the coordinate 7 and stay there.

Walk a distance of 8 clockwise, burn the tree at the coordinate 9 and stay there.

Sample Input 2

10 6
1
2
3
6
7
9

Sample Output 2

27

Sample Input 3

314159265 7
21662711
77271666
89022761
156626166
160332356
166902656
298992265

Sample Output 3

1204124749

Solution in Haskell (GHC 7.10.3) (judge verdict: Accepted)

{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TupleSections #-}
import qualified Data.ByteString.Lazy.Char8 as B
import Data.Maybe
import Data.List
import qualified Data.Vector.Unboxed as VU
-- import qualified Data.Vector.Unboxed.Mutable as VUM
-- import Control.Monad
-- import Control.Monad.ST
-- import Debug.Trace
trace _ = id

type MyS = (Int, Int, Int, Bool, Int)

solve :: Int -> Int -> [Int] -> Int
solve bigL bigN xs = max (sub vec1 vec2) (sub vec2 vec1)
  where
  vec1 = VU.fromListN bigN xs
  vec2 = VU.map (bigL -) (VU.reverse vec1)

  sub :: VU.Vector Int -> VU.Vector Int -> Int
  sub vec1 vec2 = trace ("sub " ++ show a) a
    where a = sub0 vec1 vec2
  sub0 vec1 vec2 = maximum lst
    where
    lst = trace ("lst " ++ show lst0) lst0
    lst0 = unfoldr f initState
      where f Nothing = Nothing
            f (Just st) = Just (valState st, nextState st)

    initSum = VU.sum (VU.take initLen vec1) + VU.sum (VU.take initLen vec2)
    initLen = bigN `div` 2
    initState = trace ("initState " ++ show initStateA) initStateA
    initStateA = Just (0, initLen-1, initLen-1, even bigN, initSum)
  
    nextState :: MyS -> Maybe MyS
    nextState st = trace ("nextState " ++ show (st, a)) a
      where a = nextStateA st
    nextStateA (idx1T, idx1B, idx2B, isEven, sum)
      | (not isEven) && idx2B < 0 = Nothing
      | not isEven = Just (idx1T+1, idx1B+1, idx2B, True,
                           sum - vec1 VU.! idx1T + vec1 VU.! (idx1B+1))
      | idx2B > 0 = Just (idx1T+1, idx1B, idx2B-1, False,
                           sum - vec1 VU.! idx1T - vec2 VU.! idx2B)
      | otherwise = Nothing
  
    valState :: MyS -> Int
    valState (idx1T, idx1B, idx2B, isEven, sum)
      | isEven    = 2 * sum - (min (vec1 VU.! idx1B)     (vec2 VU.! idx2B))
      | otherwise = 2 * sum + (max (vec1 VU.! (idx1B+1)) (vec2 VU.! (idx2B+1)))

readBInt :: B.ByteString -> Int
readBInt = fst . fromJust . B.readInt

tmain :: B.ByteString -> Int
tmain cont =
  let remLines0 = map B.words (B.lines cont)
      [bs_bigL,bs_bigN]:remLines1 = remLines0
      bigL = readBInt bs_bigL
      bigN = readBInt bs_bigN
      xs = map (\[x] -> readBInt x) remLines1
  in solve bigL bigN xs

outAnswer :: Int -> IO ()
outAnswer = putStrLn . show

main :: IO ()
main = outAnswer . tmain =<< B.getContents