Problem:
Score : 800 points

Problem Statement

There are N stores called Store 1, Store 2, \cdots, Store N. Takahashi, who is at his house at time 0, is planning to visit some of these stores.

It takes Takahashi one unit of time to travel from his house to one of the stores, or between any two stores.

If Takahashi reaches Store i at time t, he can do shopping there after standing in a queue for a_i \times t + b_i units of time. (We assume that it takes no time other than waiting.)

All the stores close at time T + 0.5. If Takahashi is standing in a queue for some store then, he cannot do shopping there.

Takahashi does not do shopping more than once in the same store.

Find the maximum number of times he can do shopping before time T + 0.5.

Constraints

All values in input are integers.

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

0 \leq a_i \leq 10^9

0 \leq b_i \leq 10^9

0 \leq T \leq 10^9

Input

Input is given from Standard Input in the following format:

N T
a_1 b_1
a_2 b_2
\vdots
a_N b_N

Output

Print the answer.

Sample Input 1

3 7
2 0
3 2
0 3

Sample Output 1

2

Here is one possible way to visit stores:

From time 0 to time 1: in 1 unit of time, he travels from his house to Store 1.

From time 1 to time 3: for 2 units of time, he stands in a queue for Store 1 to do shopping.

From time 3 to time 4: in 1 unit of time, he travels from Store 1 to Store 3.

From time 4 to time 7: for 3 units of time, he stands in a queue for Store 3 to do shopping.

In this way, he can do shopping twice before time 7.5.

Sample Input 2

1 3
0 3

Sample Output 2

0

Sample Input 3

5 21600
2 14
3 22
1 3
1 10
1 9

Sample Output 3

5

Sample Input 4

7 57
0 25
3 10
2 4
5 15
3 22
2 14
1 15

Sample Output 4

3

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 #-}

#define PHASE_FUSED [1]
#define PHASE_INNER [0]
#define INLINE_FUSED INLINE PHASE_FUSED
#define INLINE_INNER INLINE PHASE_INNER

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.Functor.Identity
import Data.Data
import Data.Typeable
import GHC.Generics
import System.IO
import System.IO.Unsafe (unsafeDupablePerformIO, unsafePerformIO)
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 as BSW
import qualified Data.ByteString.Char8 as BS
import qualified Data.ByteString.Lazy as BSLW
import qualified Data.ByteString.Lazy.Char8 as BSL
import qualified Data.ByteString.Builder as BSB
import qualified Data.ByteString.Unsafe as BSU
import qualified Data.ByteString.Internal as BSU
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 Unsafe.Coerce
import Foreign.ForeignPtr
import GHC.Exts (build, Int(..), Int#,
                 (+#), (*#), (-#), (<#), (>=#), (==#), quotRemInt#,
                 remInt#, uncheckedIShiftRL#, andI#, orI#,
                isTrue#, Addr#, Ptr(..))

main :: IO ()
main = do
  [n,tMax] <- map readInt . words <$> getLine
  stores <- do
    stores <- VU.unsafeThaw =<< getVecURest n (liftA2 (,) rIntL rIntL)
    vait_sortBy
      (\(a0,b0) (a1,b1) -> compare (a1*(b0+1)) (a0*(b1+1))
                           <> compare b0 b1) stores
    VU.unsafeFreeze stores
  let (nonzero,zero) = VU.span ((/=0) . fst) stores
      sumZero = VU.scanl' (+) 0 $ VU.map (+1) $ snd $ VU.unzip zero
      nonZeroRes = VU.foldl'
        (\ !prev (!a,!b) ->
           VU.zipWith min prev $ VU.cons 0 $ VU.map (\t -> (a+1)*(t+1)+b) prev)
        (VU.cons (0::Int) $ VU.replicate (floorLog2 (tMax+1)) (1+tMax))
        nonzero
  let res = VU.maximum $ (`VU.imap` nonZeroRes) $ \ !i !tEnd ->
        if tEnd > tMax then 0 else i + findIdxLeqSorted sumZero (tMax-tEnd)
  print res
  return ()

findIdxLeqSorted :: (VG.Vector v a, Ord a) => v a -> a -> Int
{-# INLINE findIdxLeqSorted #-}
findIdxLeqSorted = findIdxLeqSortedBy compare

findIdxGeqSorted :: (VG.Vector v a, Ord a) => v a -> a -> Int
{-# INLINE findIdxGeqSorted #-}
findIdxGeqSorted = findIdxGeqSortedBy compare

findIdxLeqSortedBy :: VG.Vector v a => (a -> a -> Ordering) -> v a -> a -> Int
{-# INLINE findIdxLeqSortedBy #-}
findIdxLeqSortedBy cmp vec a
  = binSearchLeq ((/=GT) . (`cmp` a) . VG.unsafeIndex vec) (-1) (VG.length vec)

findIdxGeqSortedBy :: VG.Vector v a => (a -> a -> Ordering) -> v a -> a -> Int
{-# INLINE findIdxGeqSortedBy #-}
findIdxGeqSortedBy cmp vec a
  = binSearchGeq ((/=LT) . (`cmp` a) . VG.unsafeIndex vec) (-1) (VG.length vec)


binSearchLeq,binSearchGeq :: (Int -> Bool) -> Int -> Int -> Int
{-# INLINE binSearchLeq #-}
binSearchLeq f = \le gt -> runIdentity $ binSearchLeqM (return . f) le gt
{-# INLINE binSearchGeq #-}
binSearchGeq f = \le gt -> runIdentity $ binSearchGeqM (return . f) le gt

binSearchLeqM :: (Monad m) => (Int -> m Bool) -> Int -> Int -> m Int
{-# INLINE binSearchLeqM #-}
binSearchLeqM f = go
  where
    go le gt | gt - le <= 1 = return le
             | otherwise = do
                 let !mid = le + (gt - le) `shiftR` 1
                 test <- f mid
                 if test then go mid gt else go le mid

binSearchGeqM :: (Monad m) => (Int -> m Bool) -> Int -> Int -> m Int
{-# INLINE binSearchGeqM #-}
binSearchGeqM f = go
  where
    go lt ge | ge - lt <= 1 = return ge
             | otherwise = do
                 let !mid = lt + (ge - lt) `shiftR` 1
                 test <- f mid
                 if test then go lt mid else go mid ge


floorLog2 :: Int -> Int
floorLog2 x = finiteBitSize x - countLeadingZeros x - 1
    
    
{-
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
  IL(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(readLnWith) :: StateT BS.ByteString Maybe a -> IO a
readLnWith parser = fromJust . evalStateT parser <$> BS.getLine
IL(readContentWith) :: StateT BSL.ByteString Maybe a -> IO a
readContentWith parser = fromJust . evalStateT parser <$> BSL.getContents

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

IL(rIntL) :: StateT BSL.ByteString Maybe Int
rIntL = skipSpecialL $ StateT BSL.readInt
IL(rIntS) :: StateT BS.ByteString Maybe Int
rIntS = skipSpecialS $ StateT BS.readInt
IL(rStrL) :: (MonadState BSL.ByteString m) => m BS.ByteString
rStrL = skipSpecialL $ BSL.toStrict <$> state (BSL.span (>='!'))
IL(rStrS) :: (MonadState BS.ByteString m) => m BS.ByteString
rStrS = skipSpecialS $ state $ BS.span (>='!')
IL(rCharL) :: StateT BSL.ByteString Maybe Char
rCharL = StateT BSL.uncons
IL(rCharS) :: StateT BS.ByteString Maybe Char
rCharS = StateT BS.uncons
IL(dropSpecialL) :: (MonadState BSL.ByteString m) => m ()
dropSpecialL = modify $ BSL.dropWhile (<'!')
IL(dropSpecialS) :: (MonadState BS.ByteString m) => m ()
dropSpecialS = modify $ BS.dropWhile (<'!')
IL(skipSpecialL) :: (MonadState BSL.ByteString m) => m a -> m a
skipSpecialL = (dropSpecialL *>)
IL(skipSpecialS) :: (MonadState BS.ByteString m) => m a -> m a
skipSpecialS = (dropSpecialS *>)

IL(linToMat) :: (VG.Vector v a) => Int -> Int -> v a -> V.Vector (v a)
linToMat h w lvec = vEvalElemsId $ V.generate h (\i -> VG.slice (i*w) w lvec)

IL(mLinToMat) :: (VGM.MVector v a) => Int -> Int -> v s a -> V.Vector (v s a)
mLinToMat h w lvec = vEvalElemsId $ V.generate h (\i -> VGM.slice (i*w) w lvec)
  
IL(unsafeAddrToSVec) :: Int -> Addr# -> VS.Vector Word8
unsafeAddrToSVec n addr
  = (`VS.unsafeFromForeignPtr0` n)
    $ unsafeDupablePerformIO
    $ newForeignPtr_ $ Ptr addr

IL(vEvalElemsId) :: (VG.Vector v a) => v a -> v a
vEvalElemsId = vMapFoldl (\ !_ !x -> (x,())) ()

IL(vEvalElems) :: (VG.Vector v a) => v a -> ()
vEvalElems = VG.foldl' (\ !_ !_ -> ()) () 

IL(vMapFoldl) :: (VG.Vector v b, VG.Vector v c) =>
  (a -> b -> (c,a)) -> a -> v b -> v c
vMapFoldl f a
  = VG.unstream . VFB.inplace (streamMapFoldl f a) id . VG.stream

streamMapFoldl :: (Functor m) =>
  (a -> b -> (c,a)) -> a -> VFSM.Stream m b -> VFSM.Stream m c
{-# INLINE_FUSED streamMapFoldl #-}
streamMapFoldl f a (VFSM.Stream step s) = VFSM.Stream step1 (a,s)
  where
    {-# INLINE_INNER step1 #-}
    step1 (a0,s0) =  (<$> step s0) $ \r -> case r of
      VFSM.Yield b s1 -> case f a0 b of (c,a1) -> VFSM.Yield c (a1,s1)
      VFSM.Skip    s1 -> VFSM.Skip (a0,s1)
      VFSM.Done       -> VFSM.Done

IL(svecToBS) :: VS.Vector Word8 -> BS.ByteString
svecToBS vec = BSU.fromForeignPtr ptr 0 len
  where (ptr, len) = VS.unsafeToForeignPtr0 vec


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 TS(f,a,m,init)\
  IL(f) :: forall e i s. (C(a,m) A.Ix i) => (i,i) -> init m (a i e); f
#define N(f,g,h,a,m)\
  TS(f,a,m,e->)=A.newArray;\
  TS(g,a,m,)=A.newArray_;\
  TS(h,a,m,[e]->)=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 TS

#undef IL