Problem:
Score: 400 points

Problem Statement

The Patisserie AtCoder sells cakes with number-shaped candles.
There are X, Y and Z kinds of cakes with 1-shaped, 2-shaped and 3-shaped candles, respectively.
Each cake has an integer value called deliciousness, as follows:

The deliciousness of the cakes with 1-shaped candles are A_1, A_2, ..., A_X.

The deliciousness of the cakes with 2-shaped candles are B_1, B_2, ..., B_Y.

The deliciousness of the cakes with 3-shaped candles are C_1, C_2, ..., C_Z.

Takahashi decides to buy three cakes, one for each of the three shapes of the candles, to celebrate ABC 123.

There are X \times Y \times Z such ways to choose three cakes.

We will arrange these X \times Y \times Z ways in descending order of the sum of the deliciousness of the cakes.

Print the sums of the deliciousness of the cakes for the first, second, ..., K-th ways in this list.

Constraints

1 \leq X \leq 1 \ 000

1 \leq Y \leq 1 \ 000

1 \leq Z \leq 1 \ 000

1 \leq K \leq \min(3 \ 000, X \times Y \times Z)

1 \leq A_i \leq 10 \ 000 \ 000 \ 000

1 \leq B_i \leq 10 \ 000 \ 000 \ 000

1 \leq C_i \leq 10 \ 000 \ 000 \ 000

All values in input are integers.

Input

Input is given from Standard Input in the following format:

X Y Z K
A_1 \ A_2 \ A_3 \ ... \ A_X
B_1 \ B_2 \ B_3 \ ... \ B_Y
C_1 \ C_2 \ C_3 \ ... \ C_Z

Output

Print K lines. The i-th line should contain the i-th value stated in the problem statement.

Sample Input 1

2 2 2 8
4 6
1 5
3 8

Sample Output 1

19
17
15
14
13
12
10
8

There are 2 \times 2 \times 2 = 8 ways to choose three cakes, as shown below in descending order of the sum of the deliciousness of the cakes:

(A_2, B_2, C_2): 6 + 5 + 8 = 19

(A_1, B_2, C_2): 4 + 5 + 8 = 17

(A_2, B_1, C_2): 6 + 1 + 8 = 15

(A_2, B_2, C_1): 6 + 5 + 3 = 14

(A_1, B_1, C_2): 4 + 1 + 8 = 13

(A_1, B_2, C_1): 4 + 5 + 3 = 12

(A_2, B_1, C_1): 6 + 1 + 3 = 10

(A_1, B_1, C_1): 4 + 1 + 3 = 8

Sample Input 2

3 3 3 5
1 10 100
2 20 200
1 10 100

Sample Output 2

400
310
310
301
301

There may be multiple combinations of cakes with the same sum of the deliciousness. For example, in this test case, the sum of A_1, B_3, C_3 and the sum of A_3, B_3, C_1 are both 301.
However, they are different ways of choosing cakes, so 301 occurs twice in the output.

Sample Input 3

10 10 10 20
7467038376 5724769290 292794712 2843504496 3381970101 8402252870 249131806 6310293640 6690322794 6082257488
1873977926 2576529623 1144842195 1379118507 6003234687 4925540914 3902539811 3326692703 484657758 2877436338
4975681328 8974383988 2882263257 7690203955 514305523 6679823484 4263279310 585966808 3752282379 620585736

Sample Output 3

23379871545
22444657051
22302177772
22095691512
21667941469
21366963278
21287912315
21279176669
21160477018
21085311041
21059876163
21017997739
20703329561
20702387965
20590247696
20383761436
20343962175
20254073196
20210218542
20150096547

Note that the input or output may not fit into a 32-bit integer type.

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

{-# OPTIONS_GHC -O2 #-}
{-# LANGUAGE BangPatterns          #-}
{-# LANGUAGE CPP                   #-}
{-# LANGUAGE MagicHash             #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE OverloadedStrings     #-}
{-# LANGUAGE TupleSections         #-}
{-# LANGUAGE TypeFamilies          #-}
{-# LANGUAGE ViewPatterns          #-}

import           Control.Applicative
import           Control.Exception
import           Control.Monad
import           Control.Monad.Primitive
import           Control.Monad.ST
import           Control.Monad.State.Strict
import           Data.Bool
import qualified Data.ByteString             as B
import qualified Data.ByteString.Char8       as C
import qualified Data.ByteString.Unsafe      as B
import           Data.Char
import qualified Data.Foldable               as F
import           Data.Function
import qualified Data.IntMap.Strict          as IM
import qualified Data.IntSet                 as IS
import qualified Data.List                   as L
import qualified Data.Map.Strict             as M
import           Data.Monoid
import           Data.Ord
import           Data.Primitive.MutVar
import qualified Data.Set                    as S
import           Data.Tuple
import qualified Data.Vector                 as V
import qualified Data.Vector.Generic         as G
import qualified Data.Vector.Generic.Mutable as GM
import qualified Data.Vector.Unboxed         as U
import qualified Data.Vector.Unboxed.Mutable as UM
import           Debug.Trace
import           Foreign                     hiding (void)
import           GHC.Exts
import           Unsafe.Coerce

main :: IO ()
main = do
    [x, y, z, k] <- map read.words <$> getLine :: IO [Int]
    xs <- U.unfoldrN x (C.readInt.C.dropWhile isSpace) <$> C.getLine
    ys <- U.unfoldrN y (C.readInt.C.dropWhile isSpace) <$> C.getLine
    zs <- U.unfoldrN z (C.readInt.C.dropWhile isSpace) <$> C.getLine
    putStr.unlines.map show $ solve k x y z xs ys zs


type IntIntInt = Int

data Node = Node !Int !IntIntInt deriving (Eq, Ord, Show)

encode :: Int -> Int -> Int -> IntIntInt
encode x y z = unsafeShiftL x 20 .|. unsafeShiftL y 10 .|. z

decode :: IntIntInt -> (Int, Int, Int)
decode xyz = (x, y, z)
  where
    !x = unsafeShiftR xyz 20 .&. 0x3ff
    !y = unsafeShiftR xyz 10 .&. 0x3ff
    !z = xyz .&. 0x3ff

succX :: IntIntInt -> IntIntInt
succX = (+ unsafeShiftL 1 20)

succY :: IntIntInt -> IntIntInt
succY = (+ unsafeShiftL 1 10)

succZ :: IntIntInt -> IntIntInt
succZ = (+ 1)

solve :: Int -> Int -> Int -> Int -> U.Vector Int -> U.Vector Int -> U.Vector Int -> [Int]
solve k x y z
        (U.reverse.radixSort64 -> xs)
        (U.reverse.radixSort64 -> ys)
        (U.reverse.radixSort64 -> zs)
    = take k . go IS.empty $ _HHsingleton (node $ encode 0 0 0)
  where
    priority (decode -> (i, j, k))
        | i < x, j < y, k < z = U.unsafeIndex xs i + U.unsafeIndex ys j + U.unsafeIndex zs k
        | otherwise = minBound
    node ijk = Node (priority ijk) ijk

    go !visited !set = case _HHdeleteFindMax set of
        Just (Node p ijk, set') -> p : go visited' set''
          where
            nexts = filter (flip IS.notMember visited)
                $ map ($ ijk) [succX, succY, succZ]
            visited' = F.foldl' (flip IS.insert) visited nexts
            set'' = F.foldl' (flip _HHinsert) set' $ map node nexts
        Nothing -> []

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

data MaxHeap a = MaxFork !a [MaxHeap a] | MaxEmpty

_HHempty :: MaxHeap a
_HHempty = MaxEmpty
{-# INLINE _HHempty #-}

_HHsingleton :: a -> MaxHeap a
_HHsingleton x = MaxFork x []
{-# INLINE _HHsingleton #-}

_HHnull :: MaxHeap a -> Bool
_HHnull MaxEmpty      = True
_HHnull (MaxFork _ _) = False
{-# INLINE _HHnull #-}

_HHinsert :: Ord a => a -> MaxHeap a -> MaxHeap a
_HHinsert x = _HHmerge (MaxFork x [])
{-# INLINE _HHinsert #-}

_HHMaxElem :: MaxHeap a -> Maybe a
_HHMaxElem (MaxFork x _) = Just x
_HHMaxElem MaxEmpty      = Nothing
{-# INLINE _HHMaxElem #-}

_HHdeleteMax :: Ord a => MaxHeap a -> Maybe (MaxHeap a)
_HHdeleteMax (MaxFork _ hs) = Just $ _HHmergePairs hs
_HHdeleteMax MaxEmpty       = Nothing
{-# INLINE _HHdeleteMax #-}

_HHdeleteFindMax :: Ord a => MaxHeap a -> Maybe (a, MaxHeap a)
_HHdeleteFindMax (MaxFork x hs) = Just (x, _HHmergePairs hs)
_HHdeleteFindMax MaxEmpty       = Nothing
{-# INLINE _HHdeleteFindMax #-}

_HHmerge :: Ord a => MaxHeap a -> MaxHeap a -> MaxHeap a
_HHmerge hx@(MaxFork x hxs) hy@(MaxFork y hys)
  | y <= x    = MaxFork x (hy:hxs)
  | otherwise = MaxFork y (hx:hys)
_HHmerge MaxEmpty hy = hy
_HHmerge hx _ = hx
{-# INLINE _HHmerge #-}

_HHmergePairs :: Ord a => [MaxHeap a] -> MaxHeap a
_HHmergePairs (x:y:hs) = (x <> y) <> _HHmergePairs hs
_HHmergePairs [x]      = x
_HHmergePairs []       = MaxEmpty
{-# INLINE _HHmergePairs #-}

instance Ord a => Eq (MaxHeap a) where
    (==) = (==) `on` toList

instance Ord a => Ord (MaxHeap a) where
    compare = compare `on` toList

instance Ord a => IsList (MaxHeap a) where
    type Item (MaxHeap a) = a
    fromList xs = _HHmergePairs $ map _HHsingleton xs
    toList = L.unfoldr _HHdeleteFindMax

instance (Show a, Ord a) => Show (MaxHeap a) where
    show = show . toList

instance Ord a => Monoid (MaxHeap a) where
    mempty = _HHempty
    {-# INLINE mempty #-}
    mconcat = _HHmergePairs
    {-# INLINE mconcat #-}
    mappend = _HHmerge
    {-# INLINE mappend #-}


radixSort64 :: U.Vector Int -> U.Vector Int
radixSort64 v = F.foldl' step v [0, 16, 32, 48]
  where
    mask k x = fromIntegral $ unsafeShiftR x k .&. 0xffff
    step v k = U.create $ do
        pref <- U.unsafeThaw
            . U.prescanl' (+) 0
            . U.unsafeAccumulate (+) (U.replicate 0x10000 0)
            $ U.map (flip (,) 1 . mask k) v
        res <- UM.unsafeNew $ U.length v
        U.forM_ v $ \x -> do
            let !masked = mask k x
            i <- UM.unsafeRead pref masked
            UM.unsafeWrite pref masked $ i + 1
            UM.unsafeWrite res i x
        return res
{-# INLINE radixSort64 #-}

type Parser a = C.ByteString -> Maybe (a, C.ByteString)

parseInt :: Parser Int
parseInt = C.readInt . C.dropWhile isSpace

parseInt2 :: Parser (Int, Int)
parseInt2 = runStateT $
    (,) <$> StateT (C.readInt . C.dropWhile isSpace)
        <*> StateT (C.readInt . B.unsafeTail)

parseInt3 :: Parser (Int, Int, Int)
parseInt3 = runStateT $
    (,,) <$> StateT (C.readInt . C.dropWhile isSpace)
        <*> StateT (C.readInt . B.unsafeTail)
        <*> StateT (C.readInt . B.unsafeTail)

parseInt4 :: Parser (Int, Int, Int, Int)
parseInt4 = runStateT $
    (,,,) <$> StateT (C.readInt . C.dropWhile isSpace)
        <*> StateT (C.readInt . B.unsafeTail)
        <*> StateT (C.readInt . B.unsafeTail)
        <*> StateT (C.readInt . B.unsafeTail)