Problem:
Score: 500 points

Problem Statement

N people are standing in a queue, numbered 1, 2, 3, ..., N from front to back. Each person wears a hat, which is red, blue, or green.

The person numbered i says:

"In front of me, exactly A_i people are wearing hats with the same color as mine."

Assuming that all these statements are correct, find the number of possible combinations of colors of the N people's hats.

Since the count can be enormous, compute it modulo 1000000007.

Constraints

1 \leq N \leq 100000

0 \leq A_i \leq N-1

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 A_3 ... A_N

Output

Print the number of possible combinations of colors of the N people's hats, modulo 1000000007.

Sample Input 1

6
0 1 2 3 4 5

Sample Output 1

3

We have three possible combinations, as follows:

Red, Red, Red, Red, Red, Red

Blue, Blue, Blue, Blue, Blue, Blue

Green, Green, Green, Green, Green, Green

Sample Input 2

3
0 0 0

Sample Output 2

6

Sample Input 3

54
0 0 1 0 1 2 1 2 3 2 3 3 4 4 5 4 6 5 7 8 5 6 6 7 7 8 8 9 9 10 10 11 9 12 10 13 14 11 11 12 12 13 13 14 14 15 15 15 16 16 16 17 17 17

Sample Output 3

115295190

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

{-# LANGUAGE
  ScopedTypeVariables, BangPatterns, TupleSections, ExplicitForAll,
  LambdaCase, MultiWayIf, Unsafe, RecordWildCards, FlexibleContexts, CPP,
  NoMonomorphismRestriction, GADTs, PatternGuards, MagicHash, UnboxedTuples,
  RankNTypes, EmptyDataDecls, EmptyCase, ViewPatterns, PolyKinds,
  TypeFamilies, OverloadedStrings, FlexibleInstances, UndecidableInstances,
  DefaultSignatures, GeneralizedNewtypeDeriving, StandaloneDeriving,
  DeriveGeneric, DeriveFunctor, DeriveDataTypeable, DeriveFoldable,
  DeriveTraversable, DeriveDataTypeable, FlexibleInstances,
  MultiParamTypeClasses #-}
{-# OPTIONS_GHC -O2 #-}

import Prelude
import Data.Bits
import Data.List
import Data.Maybe
import Data.Tuple
import Data.Ord
import Data.Int
import Data.Word
import Data.Char
import Data.Ratio
import Data.Function
import Data.STRef
import Data.IORef
import Data.Monoid
import Data.Functor
import Data.Data
import Data.Typeable
import GHC.Generics
import System.IO
import Control.Arrow
import Control.Applicative
import Control.Monad
import Control.Monad.Primitive
import Control.Monad.State.Strict
import Control.Monad.ST
import Control.Monad.ST.Lazy (strictToLazyST, lazyToStrictST)
import qualified Control.Monad.ST.Lazy as STL
-- import Control.Monad.ST.Safe
import Control.DeepSeq
import Data.Coerce
import qualified Data.ByteString.Char8 as BS
import qualified Data.ByteString.Lazy.Char8 as BSL
import qualified Data.ByteString.Builder as BSB
import Data.IntMap (IntMap)
import qualified Data.IntMap.Strict as IMS
import qualified Data.IntMap.Lazy as IML
import Data.IntSet (IntSet)
import qualified Data.IntSet as IS
import qualified Data.Sequence as Seq
import Data.Sequence (Seq)
import qualified Data.Array.IArray as A
import qualified Data.Array.MArray.Safe as A
import qualified Data.Array.MArray as A
import Data.Array (Array)
import Data.Array.Unboxed (UArray)
import Data.Array.IArray (IArray)
import Data.Array.MArray.Safe (MArray)
import Data.Array.IO.Safe (IOArray, IOUArray)
import Data.Array.ST.Safe (STArray, STUArray, runSTArray, runSTUArray)
import qualified Data.Vector as V
import qualified Data.Vector.Mutable as VM
import qualified Data.Vector.Unboxed as VU
import qualified Data.Vector.Unboxed.Mutable as VUM
import qualified Data.Vector.Storable as VS
import qualified Data.Vector.Storable.Mutable as VSM
import qualified Data.Vector.Primitive as VP
import qualified Data.Vector.Primitive.Mutable as VPM
import qualified Data.Vector.Generic as VG
import qualified Data.Vector.Generic.Mutable as VGM
import qualified Data.Vector.Fusion.Bundle.Monadic as VFBM
import qualified Data.Vector.Fusion.Bundle as VFB
import qualified Data.Vector.Fusion.Stream.Monadic as VFSM
import qualified Data.Vector.Fusion.Bundle.Size as VFBS
import qualified Data.Vector.Fusion.Util as VFU
import qualified Data.Attoparsec.ByteString.Char8 as Atto
import Debug.Trace
import GHC.Exts (build, Int(..), Int#,
                 (+#), (*#), (-#), (<#), (>=#), (==#), quotRemInt#,
                 remInt#, uncheckedIShiftRL#, andI#, orI#,
                isTrue#)

main :: IO ()
main = do
  n <- readInt <$> getLine
  as <- getVecULn n rIntS
  cnt <- VUM.replicate (n+1) (0::Int8)
  VUM.write cnt 0 3
  print
    =<< VU.foldM'
      (\ !acc ai -> do
          VUM.modify cnt (+1) (ai+1)
          !val <- VUM.read cnt ai
          VUM.write cnt ai (val-1)
          return $ Rem (fromIntegral val) * acc)
      (Rem 1) as
  return ()

#define MODULUS 1000000007
type RemRep = Int

modulus :: RemRep
modulus = MODULUS
-- SEE BELOW for the parameter factMax


newtype Rem = Rem {unRem :: RemRep} deriving Eq
instance Num Rem where
  Rem x - Rem y = Rem $ x-y + if x >= y then 0 else modulus
  negate (Rem x) = Rem $ modulus - x
  x + y = x - (-y)
  fromInteger x = Rem $ fromInteger (x `mod` MODULUS)
  Rem x * Rem y = Rem $ (x * y) `rem` modulus
  abs = id
  signum (Rem 0) = Rem 0
  signum _ = Rem 1

instance Show Rem where
  showsPrec = coerce (showsPrec :: Int -> RemRep -> ShowS)


newtype instance VU.Vector Rem
  = VRem { unVRem :: VU.Vector RemRep }
newtype instance VUM.MVector s Rem
  = VMRem { unVMRem :: VUM.MVector s RemRep }
instance VUM.Unbox Rem

instance VGM.MVector VUM.MVector Rem where
  {-# INLINE basicLength #-}
  basicLength = coerce `asTypeOf` (. unVMRem) $ VGM.basicLength
  {-# INLINE basicUnsafeSlice #-}
  basicUnsafeSlice
    = coerce `asTypeOf` (\f x y (VMRem v) -> VMRem $ f x y v)
      $ VGM.basicUnsafeSlice
  {-# INLINE basicOverlaps #-}
  basicOverlaps = coerce `asTypeOf` (`on` unVMRem) $ VGM.basicOverlaps
  {-# INLINE basicUnsafeNew #-}
  basicUnsafeNew = fmap (coerce `asTypeOf` VMRem) . VGM.basicUnsafeNew
  {-# INLINE basicInitialize #-}
  basicInitialize = coerce `asTypeOf` (. unVMRem) $ VGM.basicInitialize
  {-# INLINE basicUnsafeReplicate #-}
  basicUnsafeReplicate !n
    = fmap (coerce `asTypeOf` VMRem) . VGM.basicUnsafeReplicate n
    . unRem
  {-# INLINE basicUnsafeRead #-}
  basicUnsafeRead (VMRem v) !i = VGM.basicUnsafeRead v i >>=
    (return $!) . Rem
  {-# INLINE basicUnsafeWrite #-}
  basicUnsafeWrite (VMRem v) !n = VGM.basicUnsafeWrite v n . unRem
  {-# INLINE basicClear #-}
  basicClear = coerce `asTypeOf` (. unVMRem) $ VGM.basicClear
  {-# INLINE basicSet #-}
  basicSet (VMRem v) = VGM.basicSet v . unRem
  {-# INLINE basicUnsafeCopy #-}
  basicUnsafeCopy = coerce `asTypeOf` (`on` unVMRem) $ VGM.basicUnsafeCopy
  {-# INLINE basicUnsafeMove #-}
  basicUnsafeMove = coerce `asTypeOf` (`on` unVMRem) $ VGM.basicUnsafeMove
  {-# INLINE basicUnsafeGrow #-}
  basicUnsafeGrow (VMRem v) !n
    = fmap (coerce `asTypeOf` VMRem) $ VGM.basicUnsafeGrow v n

instance VG.Vector VU.Vector Rem where
  {-# INLINE basicUnsafeFreeze #-}
  basicUnsafeFreeze (VMRem v)
    = fmap (coerce `asTypeOf` VRem) $ VG.basicUnsafeFreeze v
  {-# INLINE basicUnsafeThaw #-}
  basicUnsafeThaw (VRem v)
    = fmap (coerce `asTypeOf` VMRem) $ VG.basicUnsafeThaw v
  {-# INLINE basicLength #-}
  basicLength = coerce `asTypeOf` (. unVRem) $ VG.basicLength
  {-# INLINE basicUnsafeSlice #-}
  basicUnsafeSlice
    = coerce `asTypeOf` (\f i l (VRem v) -> VRem $ f i l v)
      $ VG.basicUnsafeSlice
  {-# INLINE basicUnsafeIndexM #-}
  basicUnsafeIndexM (VRem v) !i =
    VG.basicUnsafeIndexM v i >>= (return $!) . Rem
  {-# INLINE basicUnsafeCopy #-}
  basicUnsafeCopy
    = coerce `asTypeOf` (\f (VMRem mv) (VRem v) -> f mv v)
      $ VG.basicUnsafeCopy
  {-# INLINE elemseq #-}
  elemseq = const seq


{-
The code below is the modified version of what I borrowed from the modules
Data.Vector.Algorithms.Common/Optimal/Insertion/Heap/Intro
in the package vector-algorithms-0.8.0.1 on Hackage.

The copyright notice follows:

Copyright (c) 2015 Dan Doel
Copyright (c) 2015 Tim Baumann

All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions
are met:

1. Redistributions of source code must retain the above copyright
   notice, this list of conditions and the following disclaimer.

2. Redistributions in binary form must reproduce the above copyright
   notice, this list of conditions and the following disclaimer in the
   documentation and/or other materials provided with the distribution.

3. Neither the name of the author nor the names of his contributors
   may be used to endorse or promote products derived from this software
   without specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE AUTHORS ``AS IS'' AND ANY EXPRESS OR
IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED.  IN NO EVENT SHALL THE AUTHORS OR CONTRIBUTORS BE LIABLE FOR
ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
POSSIBILITY OF SUCH DAMAGE.

------------------------------------------------------------------------------

The code in Data.Array.Vector.Algorithms.Mutable.Optimal is adapted from a C
algorithm for the same purpose. The folowing is the copyright notice for said
C code:

Copyright (c) 2004 Paul Hsieh
All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:

    Redistributions of source code must retain the above copyright notice,
    this list of conditions and the following disclaimer.

    Redistributions in binary form must reproduce the above copyright notice,
    this list of conditions and the following disclaimer in the documentation
    and/or other materials provided with the distribution.

    Neither the name of sorttest nor the names of its contributors may be
    used to endorse or promote products derived from this software without
    specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
POSSIBILITY OF SUCH DAMAGE.
-}
#define FILTER \
  | len < 2   = return ()\
  | len == 2  = vao_sort2ByOffset cmp a l\
  | len == 3  = vao_sort3ByOffset cmp a l\
  | len == 4  = vao_sort4ByOffset cmp a l
#define FILTERED FILTER | otherwise

-- | Sorts an entire array using the default ordering.
vait_sort :: (PrimMonad m, VGM.MVector v e, Ord e)
  => v (PrimState m) e -> m ()
vait_sort = vait_sortBy compare
{-# INLINABLE vait_sort #-}

-- | Sorts an entire array using a custom ordering.
vait_sortBy :: (PrimMonad m, VGM.MVector v e)
  => Comparison e -> v (PrimState m) e -> m ()
vait_sortBy cmp a = vait_sortByBounds cmp a 0 (VGM.length a)
{-# INLINE vait_sortBy #-}

-- | Sorts a portion of an array [l,u) using a custom ordering
vait_sortByBounds
  :: (PrimMonad m, VGM.MVector v e)
  => Comparison e
  -> v (PrimState m) e
  -> Int -- ^ lower index, l
  -> Int -- ^ upper index, u
  -> m ()
vait_sortByBounds cmp a l u FILTERED
  = vait_introsort cmp a (vait_ilg len) l u
 where len = u - l
{-# INLINE vait_sortByBounds #-}

-- Internal version of the introsort loop which allows partial
-- sort functions to call with a specified bound on iterations.
vait_introsort :: (PrimMonad m, VGM.MVector v e)
   => Comparison e -> v (PrimState m) e -> Int -> Int -> Int -> m ()
vait_introsort cmp a i l u = sort i l u >> vais_sortByBounds cmp a l u
 where
 sort 0 l u = vah_sortByBounds cmp a l u
 sort d l u
   | len < vait_threshold = return ()
   | otherwise = do
       vao_sort3ByIndex cmp a c l (u-1)
        -- sort the median into the lowest position
       p <- VGM.unsafeRead a l
       mid <- vait_partitionBy cmp a p (l+1) u
       VGM.unsafeSwap a l (mid - 1)
       sort (d-1) mid u
       sort (d-1) l   (mid - 1)
  where
    len = u - l
    c   = vac_midPoint u l
{-# INLINE vait_introsort #-}

vait_partitionBy :: forall m v e. (PrimMonad m, VGM.MVector v e)
  => Comparison e -> v (PrimState m) e -> e -> Int -> Int -> m Int
vait_partitionBy cmp a = partUp
 where
 partUp :: e -> Int -> Int -> m Int
 partUp p l u
   | l < u = do
       e <- VGM.unsafeRead a l
       case cmp e p of
         LT -> partUp p (l+1) u
         _  -> partDown p l (u-1)
   | otherwise = return l

 partDown :: e -> Int -> Int -> m Int
 partDown p l u
   | l < u = do
       e <- VGM.unsafeRead a u
       case cmp p e of
         LT -> partDown p l (u-1)
         _  -> VGM.unsafeSwap a l u >> partUp p (l+1) u
   | otherwise = return l
{-# INLINE vait_partitionBy #-}

vait_ilg :: Int -> Int
vait_ilg !m = 2 * (finiteBitSize m - countLeadingZeros m - 1)

vait_threshold :: Int
vait_threshold = 18
{-# INLINE vait_threshold #-}

vah_sortByBounds :: (PrimMonad m, VGM.MVector v e)
  => Comparison e -> v (PrimState m) e -> Int -> Int -> m ()
vah_sortByBounds cmp a l u FILTERED
  = vah_heapify cmp a l u >> vah_sortHeap cmp a l (l+4) u
    >> vao_sort4ByOffset cmp a l
 where len = u - l
{-# INLINE vah_sortByBounds #-}

vah_heapify :: (PrimMonad m, VGM.MVector v e) => Comparison e
  -> v (PrimState m) e -> Int -> Int -> m ()
vah_heapify cmp a l u = loop $ (len - 1) `shiftR` 2
  where
    len = u - l
    loop k | k < 0 = return ()
    loop k = VGM.unsafeRead a (l+k) >>= \e ->
      vah_siftByOffset cmp a e l k len >> loop (k - 1)
{-# INLINE vah_heapify #-}

vah_pop :: (PrimMonad m, VGM.MVector v e)
  => Comparison e -> v (PrimState m) e -> Int -> Int -> m ()
vah_pop cmp a l u = vah_popTo cmp a l u u
{-# INLINE vah_pop #-}

vah_popTo :: (PrimMonad m, VGM.MVector v e) => Comparison e
  -> v (PrimState m) e -> Int -> Int -> Int -> m ()
vah_popTo cmp a l u t = do
  al <- VGM.unsafeRead a l
  at <- VGM.unsafeRead a t
  VGM.unsafeWrite a t al
  vah_siftByOffset cmp a at l 0 (u - l)
{-# INLINE vah_popTo #-}

vah_sortHeap :: (PrimMonad m, VGM.MVector v e) => Comparison e
  -> v (PrimState m) e -> Int -> Int -> Int -> m ()
vah_sortHeap cmp a l m u = loop (u-1) >> VGM.unsafeSwap a l m
 where
 loop k
   | m < k     = vah_pop cmp a l k >> loop (k-1)
   | otherwise = return ()
{-# INLINE vah_sortHeap #-}

vah_siftByOffset :: (PrimMonad m, VGM.MVector v e)
  => Comparison e -> v (PrimState m) e -> e -> Int -> Int -> Int -> m ()
vah_siftByOffset cmp a val off start len = sift val start len
 where
 sift val root len
   | child < len = do
       (child', ac) <- vah_maximumChild cmp a off child len
       case cmp val ac of
         LT -> VGM.unsafeWrite a (root + off) ac >> sift val child' len
         _  -> VGM.unsafeWrite a (root + off) val
   | otherwise = VGM.unsafeWrite a (root + off) val
  where child = root `shiftL` 2 + 1
{-# INLINE vah_siftByOffset #-}

vah_maximumChild :: (PrimMonad m, VGM.MVector v e)
  => Comparison e -> v (PrimState m) e -> Int -> Int -> Int -> m (Int,  e)
vah_maximumChild cmp a off child1 len
  | child4 < len = do
      ac1 <- VGM.unsafeRead a (child1 + off)
      ac2 <- VGM.unsafeRead a (child2 + off)
      ac3 <- VGM.unsafeRead a (child3 + off)
      ac4 <- VGM.unsafeRead a (child4 + off)
      return $ case cmp ac1 ac2 of
        LT -> case cmp ac2 ac3 of
          LT -> case cmp ac3 ac4 of
            LT -> (child4, ac4)
            _  -> (child3, ac3)
          _  -> case cmp ac2 ac4 of
            LT -> (child4, ac4)
            _  -> (child2, ac2)
        _  -> case cmp ac1 ac3 of
          LT -> case cmp ac3 ac4 of
            LT -> (child4, ac4)
            _  -> (child3, ac3)
          _  -> case cmp ac1 ac4 of
            LT -> (child4, ac4)
            _  -> (child1, ac1)
  | child3 < len = do
      ac1 <- VGM.unsafeRead a (child1 + off)
      ac2 <- VGM.unsafeRead a (child2 + off)
      ac3 <- VGM.unsafeRead a (child3 + off)
      return $ case cmp ac1 ac2 of
        LT -> case cmp ac2 ac3 of
          LT -> (child3, ac3)
          _  -> (child2, ac2)
        _  -> case cmp ac1 ac3 of
          LT -> (child3, ac3)
          _  -> (child1, ac1)
  | child2 < len = do
      ac1 <- VGM.unsafeRead a (child1 + off)
      ac2 <- VGM.unsafeRead a (child2 + off)
      return $ case cmp ac1 ac2 of
        LT -> (child2, ac2)
        _  -> (child1, ac1)
  | otherwise    = do
      ac1 <- VGM.unsafeRead a (child1 + off); return (child1, ac1)
 where
 child2 = child1 + 1
 child3 = child1 + 2
 child4 = child1 + 3
{-# INLINE vah_maximumChild #-}

vais_sortByBounds :: (PrimMonad m, VGM.MVector v e)
  => Comparison e -> v (PrimState m) e -> Int -> Int -> m ()
vais_sortByBounds cmp a l u FILTERED
  = vao_sort4ByOffset cmp a l >> vais_sortByBounds' cmp a l (l + 4) u
 where len = u - l
{-# INLINE vais_sortByBounds #-}

vais_sortByBounds' :: (PrimMonad m, VGM.MVector v e)
  => Comparison e -> v (PrimState m) e -> Int -> Int -> Int -> m ()
vais_sortByBounds' cmp a l m u = sort m
 where
 sort i
   | i < u     = do
       v <- VGM.unsafeRead a i
       vais_insert cmp a l v i
       sort (i+1)
   | otherwise = return ()
{-# INLINE vais_sortByBounds' #-}

vais_insert :: (PrimMonad m, VGM.MVector v e)
  => Comparison e -> v (PrimState m) e -> Int -> e -> Int -> m ()
vais_insert cmp a l = loop
 where
 loop val j
   | j <= l    = VGM.unsafeWrite a l val
   | otherwise = do
       e <- VGM.unsafeRead a (j - 1)
       case cmp val e of
         LT -> VGM.unsafeWrite a j e >> loop val (j - 1)
         _  -> VGM.unsafeWrite a j val
{-# INLINE vais_insert #-}

#define vec_writes2(a,v,w)\
  (VGM.unsafeWrite a v >> VGM.unsafeWrite a w)
#define vec_writes3(a,v,w,x)\
  (VGM.unsafeWrite a v >> vec_writes2(a,w,x))
#define vec_writes4(a,v,w,x,y)\
  (VGM.unsafeWrite a v >> vec_writes3(a,w,x,y))

vao_sort2ByOffset :: (PrimMonad m, VGM.MVector v e)
  => Comparison e -> v (PrimState m) e -> Int -> m ()
vao_sort2ByOffset cmp a off = vao_sort2ByIndex cmp a off (off + 1)
{-# INLINABLE vao_sort2ByOffset #-}

vao_sort2ByIndex :: (PrimMonad m, VGM.MVector v e)
  => Comparison e -> v (PrimState m) e -> Int -> Int -> m ()
vao_sort2ByIndex cmp a i j = do
  a0 <- VGM.unsafeRead a i
  a1 <- VGM.unsafeRead a j
  case cmp a0 a1 of
    GT -> vec_writes2(a,i a1,j a0)
    _  -> return ()
{-# INLINABLE vao_sort2ByIndex #-}

vao_sort3ByOffset :: (PrimMonad m, VGM.MVector v e)
  => Comparison e -> v (PrimState m) e -> Int -> m ()
vao_sort3ByOffset cmp a off = vao_sort3ByIndex cmp a off (off + 1) (off + 2)
{-# INLINABLE vao_sort3ByOffset #-}

vao_sort3ByIndex :: (PrimMonad m, VGM.MVector v e)
  => Comparison e -> v (PrimState m) e -> Int -> Int -> Int -> m ()
vao_sort3ByIndex cmp a i j k = do
  a0 <- VGM.unsafeRead a i
  a1 <- VGM.unsafeRead a j
  a2 <- VGM.unsafeRead a k
  case cmp a0 a1 of
    GT -> case cmp a0 a2 of
      GT -> case cmp a2 a1 of
        LT -> vec_writes2(a,i a2,{-j a1-}k a0)
        _  -> vec_writes3(a,i a1,j a2,k a0)
      _  -> vec_writes2(a,i a1,j a0{-k a2-})
    _  -> case cmp a1 a2 of
      GT -> case cmp a0 a2 of
        GT -> vec_writes3(a,i a2,j a0,k a1)
        _  -> vec_writes2(a,{-i a0-}j a2,k a1)
      _  -> return ()
{-# INLINABLE vao_sort3ByIndex #-}

vao_sort4ByOffset :: (PrimMonad m, VGM.MVector v e)
  => Comparison e -> v (PrimState m) e -> Int -> m ()
vao_sort4ByOffset cmp a off
  = vao_sort4ByIndex cmp a off (off + 1) (off + 2) (off + 3)
{-# INLINABLE vao_sort4ByOffset #-}

vao_sort4ByIndex :: (PrimMonad m, VGM.MVector v e)
  => Comparison e -> v (PrimState m) e -> Int -> Int -> Int -> Int -> m ()
vao_sort4ByIndex cmp a i j k l = do
  a0 <- VGM.unsafeRead a i
  a1 <- VGM.unsafeRead a j
  a2 <- VGM.unsafeRead a k
  a3 <- VGM.unsafeRead a l
  case cmp a0 a1 of
    GT -> case cmp a0 a2 of
      GT -> case cmp a1 a2 of
        GT -> case cmp a1 a3 of
          GT -> case cmp a2 a3 of
            GT -> vec_writes4(a,i a3,j a2,k a1,l a0)
            _  -> vec_writes4(a,i a2,j a3,k a1,l a0)
          _  -> case cmp a0 a3 of
            GT -> vec_writes3(a,i a2,{-j a1-}k a3,l a0)
            _  -> vec_writes2(a,i a2,{-j a1-}k a0{-l a3-})
        _ -> case cmp a2 a3 of
          GT -> case cmp a1 a3 of
            GT -> vec_writes2(a,i a3,{-j a1;k a2-}l a0)
            _  -> vec_writes3(a,i a1,j a3,{-k a2-}l a0)
          _  -> case cmp a0 a3 of
            GT -> vec_writes4(a,i a1,j a2,k a3,l a0)
            _  -> vec_writes3(a,i a1,j a2,k a0{-l a3-})
      _  -> case cmp a0 a3 of
        GT -> case cmp a1 a3 of
          GT -> vec_writes3(a,i a3,{-j a1-}k a0,l a2)
          _  -> vec_writes4(a,i a1,j a3,k a0,l a2)
        _  -> case cmp a2 a3 of
          GT -> vec_writes4(a,i a1,j a0,k a3,l a2)
          _  -> vec_writes2(a,i a1,j a0{-k a2;l a3-})
    _  -> case cmp a1 a2 of
      GT -> case cmp a0 a2 of
        GT -> case cmp a0 a3 of
          GT -> case cmp a2 a3 of
            GT -> vec_writes4(a,i a3,j a2,k a0,l a1)
            _  -> vec_writes4(a,i a2,j a3,k a0,l a1)
          _  -> case cmp a1 a3 of
            GT -> vec_writes4(a,i a2,j a0,k a3,l a1)
            _  -> vec_writes3(a,i a2,j a0,k a1{-l a3-})
        _  -> case cmp a2 a3 of
          GT -> case cmp a0 a3 of
            GT -> vec_writes3(a,i a3,j a0,{-k a2-}l a1)
            _  -> vec_writes2(a,{-i a0-}j a3,{-k a2-}l a1)
          _  -> case cmp a1 a3 of
            GT -> vec_writes3(a,{-i a0-}j a2,k a3,l a1)
            _  -> vec_writes2(a,{-i a0-}j a2,k a1{-l a3-})
      _  -> case cmp a1 a3 of
        GT -> case cmp a0 a3 of
          GT -> vec_writes4(a,i a3,j a0,k a1,l a2)
          _  -> vec_writes3(a,{-i a0-}j a3,k a1,l a2)
        _  -> case cmp a2 a3 of
          GT -> vec_writes2(a,{-i a0;j a1-}k a3,l a2)
          _  -> return () {-i a0;j a1;k a2;l a3-}
{-# INLINABLE vao_sort4ByIndex #-}

-- | A type of comparisons between two values of a given type.
type Comparison e = e -> e -> Ordering

vac_midPoint :: Int -> Int -> Int
vac_midPoint a b =
  fromIntegral $ (toWord a + toWord b) `div` 2
  where
    toWord :: Int -> Word
    toWord = fromIntegral
{-# INLINE vac_midPoint #-}

#undef vec_writes2
#undef vec_writes3
#undef vec_writes4
#undef FILTERED
#undef FILTER

{- Borrowed code end -}


#define IL(f) {-# INLINE f #-}; f

IL(putBuilder) = BSB.hPutBuilder stdout

printVecInLines, printVecInSpcSepLn ::
  (VG.Vector v a, ShowAsBuilder a) => v a -> IO ()
IL(printVecInLines) = putBuilder . v2BLines
IL(printVecInSpcSepLn) = putBuilder . v2BSpcSepLn

class ShowAsBuilder a where
  showAsBuilder :: a -> BSB.Builder
  default showAsBuilder :: (Show a) => a -> BSB.Builder
  showAsBuilder = BSB.string8 . show

-- Inconsistent with show
instance (ShowAsBuilder a, VG.Vector v a) => ShowAsBuilder (v a) where
  IL(showAsBuilder) = v2BSpcSep

#define INS(t,f) instance ShowAsBuilder t where { IL(showAsBuilder)=f }
INS(Int,BSB.intDec)
INS(Int8,BSB.int8Dec)
INS(Int16,BSB.int16Dec)
INS(Int32,BSB.int32Dec)
INS(Int64,BSB.int64Dec)
INS(Word,BSB.wordDec)
INS(Word8,BSB.word8Dec)
INS(Word16,BSB.word16Dec)
INS(Word32,BSB.word32Dec)
INS(Word64,BSB.word64Dec)
INS(Integer,BSB.integerDec)
INS(Float,BSB.floatDec)
INS(Double,BSB.doubleDec)
-- INS(String,BSB.string8) -- Inconsistent with Show
-- INS(BS.ByteString,BSB.byteString) -- Inconsistent with Show
-- INS(BSL.ByteString,BSB.lazyByteString) -- Inconsisitent with Show
#undef INS

-- Inconsistent with Show
instance (ShowAsBuilder a, ShowAsBuilder b) => ShowAsBuilder (a,b) where
  IL(showAsBuilder) = showTupAsBuilder
instance (ShowAsBuilder a, ShowAsBuilder b, ShowAsBuilder c) =>
  ShowAsBuilder (a,b,c) where
  IL(showAsBuilder) = showTup3AsBuilder
instance (ShowAsBuilder a, ShowAsBuilder b, ShowAsBuilder c, ShowAsBuilder d) =>
  ShowAsBuilder (a,b,c,d) where
  IL(showAsBuilder) = showTup4AsBuilder

IL(showTupAsBuilderWith)
  :: (a -> BSB.Builder) -> (b -> BSB.Builder) -> (a,b) -> BSB.Builder
showTupAsBuilderWith showA showB
  = \(a,b) -> (showA a <>) $ BSB.char7 ' ' <> showB b
IL(showTupAsBuilder) :: (ShowAsBuilder a, ShowAsBuilder b)
  => (a,b) -> BSB.Builder
showTupAsBuilder = showTupAsBuilderWith showAsBuilder showAsBuilder 

IL(showTup3AsBuilderWith) :: (a -> BSB.Builder) -> (b -> BSB.Builder) ->
  (c -> BSB.Builder) -> (a,b,c) -> BSB.Builder
showTup3AsBuilderWith showA showB showC
  = \(a,b,c) -> (showA a <>) $ (BSB.char7 ' ' <>) $ (showB b <>)
                $ (BSB.char7 ' ' <>) $ showC c
IL(showTup3AsBuilder) :: (ShowAsBuilder a, ShowAsBuilder b, ShowAsBuilder c)
  => (a,b,c) -> BSB.Builder
showTup3AsBuilder
  = showTup3AsBuilderWith showAsBuilder showAsBuilder showAsBuilder

IL(showTup4AsBuilderWith) :: (a -> BSB.Builder) -> (b -> BSB.Builder) ->
  (c -> BSB.Builder) -> (d -> BSB.Builder) -> (a,b,c,d) -> BSB.Builder
showTup4AsBuilderWith showA showB showC showD
  = \(a,b,c,d) -> (showA a <>) $ (BSB.char7 ' ' <>)
                  $ showTup3AsBuilderWith showB showC showD (b,c,d)
IL(showTup4AsBuilder) ::
  (ShowAsBuilder a, ShowAsBuilder b, ShowAsBuilder c, ShowAsBuilder d) =>
  (a,b,c,d) -> BSB.Builder
showTup4AsBuilder = showTup4AsBuilderWith showAsBuilder showAsBuilder
                    showAsBuilder showAsBuilder

v2BSpcSepLn, v2BSpcSep, v2BConcat, v2BLines ::
  (VG.Vector v a, ShowAsBuilder a)
  => v a -> BSB.Builder
IL(v2BSpcSepLn) = v2BSpcSepLnWith showAsBuilder
IL(v2BSpcSep) = v2BSpcSepWith showAsBuilder
IL(v2BConcat) = v2BConcatWith showAsBuilder
IL(v2BLines) = v2BLinesWith showAsBuilder


v2BSpcSepLnWith, v2BSpcSepWith, v2BConcatWith, v2BLinesWith ::
  (VG.Vector v a)
  => (a -> BSB.Builder) -- ^ show function
  -> v a -> BSB.Builder
IL(v2BSpcSepLnWith) = v2BSpcSepPostfWith $ BS.singleton '\n'
IL(v2BSpcSepWith) = v2BSpcSepPostfWith BS.empty
IL(v2BConcatWith) showFct = VG.foldr ((<>) . showFct) mempty
IL(v2BLinesWith) showFct
  = VG.foldr (\ a -> (showFct a <>) . (BSB.char7 '\n' <>)) mempty


v2BSpcSepPostf :: (VG.Vector v a, ShowAsBuilder a)
  => BS.ByteString -- ^ postfix
  -> v a -> BSB.Builder
IL(v2BSpcSepPostf) = (`v2BSpcSepPostfWith` showAsBuilder)

v2BSpcSepPostfWith :: (VG.Vector v a)
  => BS.ByteString -- ^ postfix
  -> (a -> BSB.Builder) -- ^ show function
  -> v a -> BSB.Builder
IL(v2BSpcSepPostfWith) = vecToBuilder BS.empty $ BS.singleton ' '

IL(vecToBuilder) :: (VG.Vector v a)
  => BS.ByteString -- ^ prefix
  -> BS.ByteString -- ^ separator
  -> BS.ByteString -- ^ postfix
  -> (a -> BSB.Builder) -- ^ show function
  -> v a -> BSB.Builder
vecToBuilder !prefix !separator !postfix
  = vecToBuilder_ (BSB.byteString prefix)
                  (BSB.byteString separator)
                  (BSB.byteString postfix)


IL(vecToBuilder_) :: (VG.Vector v a)
  => BSB.Builder -- ^ prefix
  -> BSB.Builder -- ^ separator
  -> BSB.Builder -- ^ postfix
  -> (a -> BSB.Builder) -- ^ show function
  -> v a -> BSB.Builder
vecToBuilder_ !prefix !separator !postfix showFct = \vec -> prefix <>
  VG.foldr
  (\ a rest !prefx -> prefx <> (showFct a <> rest separator))
  (const postfix) vec mempty

IL(evalVals) :: [a] -> [a]
evalVals xs = build $ \c n -> foldr (c $!) n xs
IL(forceVals) :: (NFData a) => [a] -> [a]
forceVals xs = build $ \c n -> foldr (c $!!) n xs

IL(getVecGLn) :: (VG.Vector v a) =>
  Int -> StateT BS.ByteString Maybe a -> IO (v a)
getVecGLn n s = VG.unfoldrN n (runStateT s) <$> BS.getLine
IL(getVecGRest) :: (VG.Vector v a) =>
  Int -> StateT BSL.ByteString Maybe a -> IO (v a)
getVecGRest n s = VG.unfoldrN n (runStateT s) <$> BSL.getContents
IL(getVecLn) :: Int -> StateT BS.ByteString Maybe a -> IO (V.Vector a)
getVecLn = getVecGLn
IL(getVecRest) :: Int -> StateT BSL.ByteString Maybe a -> IO (V.Vector a)
getVecRest = getVecGRest
IL(getVecULn) :: (VU.Unbox a) =>
  Int -> StateT BS.ByteString Maybe a -> IO (VU.Vector a)
getVecULn = getVecGLn
IL(getVecURest) :: (VU.Unbox a) =>
  Int -> StateT BSL.ByteString Maybe a -> IO (VU.Vector a)
getVecURest = getVecGRest

rInt :: StateT BSL.ByteString Maybe Int
IL(rInt) = StateT $ BSL.readInt . BSL.dropWhile (<'!')
rIntS :: StateT BS.ByteString Maybe Int
IL(rIntS) = StateT $ BS.readInt . BS.dropWhile (<'!')

unlessM, whenM :: (Monad m) => m Bool -> m () -> m ()
IL(whenM) = (. flip when) . (>>=)
IL(unlessM) = (. flip unless) . (>>=)

IL(wrA) = A.writeArray
IL(rdA) = A.readArray
IL(mdA) = \arr f !i -> do
  ai <- rdA arr i
  let fai = f ai 
  wrA arr i fai
  return (ai,fai)
{-# INLINE mdA' #-}
mdA' = \arr f !i -> do
  !ai <- rdA arr i
  let !fai = f ai
  wrA arr i fai
  return (ai,fai)
IL(swapA) = \arr !i !j -> do
  ai <- rdA arr i
  wrA arr i =<< rdA arr j
  wrA arr j ai

#define D(f,r,d)\
  IL(f) :: Integral a=>a->d; f=fromIntegral;\
  IL(r) :: String->d; r=read
#define C(f,r,g,h,d) D(f,r,d);\
  g,h :: RealFrac a=>a->d; IL(g)=floor; IL(h)=ceiling
C(_toInteger_,readInteger,floorInteger,ceilInteger,Integer)
C(toInt,readInt,floorInt,ceilInt,Int)
C(toI8,readI8,floorI8,ceilI8,Int8)
C(toI16,readI16,floorI16,ceilI16,Int16)
C(toI32,readI32,floorI32,ceilI32,Int32)
C(toI64,readI64,floorI64,ceilI64,Int64)
C(toWord,readWord,floorWord,ceilWord,Word)
C(toW8,readW8,floorW8,ceilW8,Word8)
C(toW16,readW16,floorW16,ceilW16,Word16)
C(toW32,readW32,floorW32,ceilW32,Word32)
C(toW64,readW64,floorW64,ceilW64,Word64)
D(toDouble,readDouble,Double)
D(toFloat,readFloat,Float)
#undef D
#undef C

#define N(f,g,h,a,m)\
  IL(f) :: forall e i s. (C(a,m)A.Ix i) => (i,i) -> e -> m (a i e);\
  f=A.newArray;\
  IL(g) :: forall e i s. (C(a,m)A.Ix i) => (i,i) -> m (a i e);\
  g=A.newArray_;\
  IL(h) :: forall e i s. (C(a,m)A.Ix i) => (i,i) -> [e] -> m (a i e);\
  h=A.newListArray
#define C(a,m)
N(newIOA,newIOA_,newIOAL,IOArray,IO)
N(newSTA,newSTA_,newSTAL,STArray s,ST s)
#undef C
#define C(a,m) MArray (a) e (m), 
N(newIOUA,newIOUA_,newIOUAL,IOUArray,IO)
N(newSTUA,newSTUA_,newSTUAL,STUArray s,ST s)
#undef C
#undef N

#undef IL