Problem:
Score : 600 points

Problem Statement

10^9 contestants, numbered 1 to 10^9, will compete in a competition.
There will be two contests in this competition.

The organizer prepared N problems, numbered 1 to N, to use in these contests.
When Problem i is presented in a contest, it will be solved by all contestants from Contestant L_i to Contestant R_i (inclusive), and will not be solved by any other contestants.

The organizer will use these N problems in the two contests.
Each problem must be used in exactly one of the contests, and each contest must have at least one problem.

The joyfulness of each contest is the number of contestants who will solve all the problems in the contest.
Find the maximum possible total joyfulness of the two contests.

Constraints

2 \leq N \leq 10^5

1 \leq L_i \leq R_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
L_1 R_1
L_2 R_2
\vdots
L_N R_N

Output

Print the maximum possible total joyfulness of the two contests.

Sample Input 1

4
4 7
1 4
5 8
2 5

Sample Output 1

6

The optimal choice is:

Use Problem 1 and 3 in the first contest. Contestant 5, 6, and 7 will solve both of them, so the joyfulness of this contest is 3.

Use Problem 2 and 4 in the second contest. Contestant 2, 3, and 4 will solve both of them, so the joyfulness of this contest is 3.

The total joyfulness of these two contests is 6. We cannot make the total joyfulness greater than 6.

Sample Input 2

4
1 20
2 19
3 18
4 17

Sample Output 2

34

Sample Input 3

10
457835016 996058008
456475528 529149798
455108441 512701454
455817105 523506955
457368248 814532746
455073228 459494089
456651538 774276744
457667152 974637457
457293701 800549465
456580262 636471526

Sample Output 3

540049931

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

import Data.List (sortOn)
import Data.Tuple
import qualified Data.Ord as Ord
import qualified Data.Set as Set
import Control.Monad.Writer.Strict

newtype DList a = DL { unDL :: [a] -> [a] }

dListFromList :: [a] -> DList a
dListFromList = DL . (++)
dListToList :: DList a -> [a]
dListToList = ($ []) . unDL
dListEmpty :: DList a
dListEmpty = DL id
dListSingleton :: a -> DList a
dListSingleton = DL . (:)
dListAppend :: DList a -> DList a -> DList a
dListAppend xs ys = DL (unDL xs . unDL ys)
instance Monoid (DList a) where
	mempty = dListEmpty
	mappend = dListAppend

list2Tuple (a:b:_) = (a, b)

data RangeTree = RangeTree {
	startpoint :: Set.Set Int,
	endpoint :: Set.Set Int
} deriving (Show)

emptyTree :: RangeTree
emptyTree = RangeTree Set.empty Set.empty

null :: RangeTree -> Bool
null tree = Set.null $ startpoint tree

length :: RangeTree -> Int
length t = max 0 length'
	where
		length' = ep - sp + 1
		sp = Set.findMax $ startpoint t
		ep = Set.findMin $ endpoint t

insert :: (Int, Int) -> RangeTree -> RangeTree
insert (l, r) tree = RangeTree newSp newEp
	where
		newSp = Set.insert l $ startpoint tree
		newEp = Set.insert r $ endpoint tree

delete ::  (Int, Int) -> RangeTree -> RangeTree
delete (l, r) tree = RangeTree newSp newEp
	where
		newSp = Set.delete l $ startpoint tree
		newEp = Set.delete r $ endpoint tree

fromList :: [(Int, Int)] -> RangeTree
fromList = foldr Main.insert emptyTree

move :: (Int, Int) -> Writer (DList Int) (RangeTree, RangeTree) -> Writer (DList Int) (RangeTree, RangeTree)
move (l, r) trees = do
	(tree1, tree2) <- trees
	let newTree1 = delete (l, r) tree1
	let newTree2 = insert (l, r) tree2
	when (not (Main.null newTree1) && not (Main.null newTree2)) $ do
		tell $ dListSingleton $ Main.length newTree1 + Main.length newTree2
		return ()
	return (newTree1, newTree2)

simple :: [(Int, Int)] -> Int
simple lrs = longest + shortest
	where
		shortest = max 0 shortest'
		shortest' = maximum ls - minimum rs
		longest = maximum $ map (uncurry (-) . swap) lrs
		ls = map fst lrs
		rs = map snd lrs

main :: IO ()
main = do
	n <- readLn
	lrs <- sortOn fst . map (list2Tuple . map read . words) . take n . lines <$> getContents :: IO [(Int, Int)]
	let initTree = writer ((fromList lrs, emptyTree), dListEmpty)
	let lengths = execWriter $ foldr move initTree lrs
	let cand1 = maximum $ dListToList lengths
	let cand2 = simple lrs
	print $ max cand1 cand2