Problem:
Score : 400 points

Problem Statement

We have a board with a 2 \times N grid.
Snuke covered the board with N dominoes without overlaps.
Here, a domino can cover a 1 \times 2 or 2 \times 1 square.

Then, Snuke decided to paint these dominoes using three colors: red, cyan and green.
Two dominoes that are adjacent by side should be painted by different colors.
Here, it is not always necessary to use all three colors.

Find the number of such ways to paint the dominoes, modulo 1000000007.

The arrangement of the dominoes is given to you as two strings S_1 and S_2 in the following manner:

Each domino is represented by a different English letter (lowercase or uppercase).

The j-th character in S_i represents the domino that occupies the square at the i-th row from the top and j-th column from the left.

Constraints

1 \leq N \leq 52

|S_1| = |S_2| = N

S_1 and S_2 consist of lowercase and uppercase English letters.

S_1 and S_2 represent a valid arrangement of dominoes.

Input

Input is given from Standard Input in the following format:

N
S_1
S_2

Output

Print the number of such ways to paint the dominoes, modulo 1000000007.

Sample Input 1

3
aab
ccb

Sample Output 1

6

There are six ways as shown below:

Sample Input 2

1
Z
Z

Sample Output 2

3

Note that it is not always necessary to use all the colors.

Sample Input 3

52
RvvttdWIyyPPQFFZZssffEEkkaSSDKqcibbeYrhAljCCGGJppHHn
RLLwwdWIxxNNQUUXXVVMMooBBaggDKqcimmeYrhAljOOTTJuuzzn

Sample Output 3

958681902

Solution in Haskell (GHC 8.8.3) (judge verdict: Accepted)

{-# OPTIONS_GHC -O2 #-}
{-# LANGUAGE BangPatterns, BinaryLiterals, CPP, DerivingStrategies  #-}
{-# LANGUAGE DerivingVia, FlexibleContexts, FlexibleInstances       #-}
{-# LANGUAGE GeneralizedNewtypeDeriving, KindSignatures, LambdaCase #-}
{-# LANGUAGE MagicHash, MultiParamTypeClasses, MultiWayIf           #-}
{-# LANGUAGE NumericUnderscores, OverloadedStrings, PatternSynonyms #-}
{-# LANGUAGE RankNTypes, RecordWildCards, ScopedTypeVariables       #-}
{-# LANGUAGE StandaloneDeriving, TupleSections, TypeApplications    #-}
{-# LANGUAGE TypeFamilies, TypeInType, UnboxedTuples, ViewPatterns  #-}

import           Control.Applicative
import           Control.Exception
import           Control.Monad
import           Control.Monad.Primitive
import           Control.Monad.Reader
import           Control.Monad.ST
import           Control.Monad.State.Strict
import           Data.Bifunctor
import           Data.Bool
import qualified Data.ByteString                   as B
import qualified Data.ByteString.Builder           as B
import qualified Data.ByteString.Char8             as C
import qualified Data.ByteString.Internal          as B
import qualified Data.ByteString.Unsafe            as B
import           Data.Char
import qualified Data.Foldable                     as F
import           Data.Function
import           Data.Functor.Identity
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
import           Data.Proxy
import           Data.Ratio
import           Data.Semigroup
import qualified Data.Set                          as S
import           Data.Tuple
import qualified Data.Vector                       as V
import qualified Data.Vector.Algorithms.Intro      as Intro
import qualified Data.Vector.Fusion.Stream.Monadic as MS
import qualified Data.Vector.Fusion.Bundle.Monadic as MB
import           Data.Vector.Fusion.Util
import qualified Data.Vector.Generic               as G
import qualified Data.Vector.Generic.Mutable       as GM
import qualified Data.Vector.Mutable               as VM
import qualified Data.Vector.Primitive             as P
import qualified Data.Vector.Primitive.Mutable     as PM
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           GHC.TypeLits
import           System.IO
import           Unsafe.Coerce

#define MOD 1000000007

main :: IO ()
main = do
    n <- readLn @Int
    s1 <- getLine
    s2 <- getLine
    print $ solve n s1 s2

parse :: String -> [Bool]
parse (x:y:s)
    | x == y = True : parse s
    | otherwise = False : parse (y:s)
parse [_] = [False]
parse [] = []


solve :: Int -> String -> String -> IntMod
solve n (parse -> cs0) _ = case cs0 of
    (True:_) -> go 6 cs0
    (False:_) -> go 3 cs0
  where
    go !acc (True:True:cs) = go (3*acc) (True:cs)
    go !acc (True:False:cs) = go acc (False:cs)
    go !acc (False:True:cs) = go (2*acc) (True:cs)
    go !acc (False:False:cs) = go (2*acc) (False:cs)
    go acc _ = acc

-------------------------------------------------------------------------------
-- Utils
-------------------------------------------------------------------------------
rep :: (Monad m) => Int -> (Int -> m ()) -> m ()
rep n = flip MS.mapM_ (stream 0 n)
{-# INLINE rep #-}
rev :: (Monad m) => Int -> (Int -> m ()) -> m ()
rev !n = flip MS.mapM_ (streamR 0 n)
{-# INLINE rev #-}
stream :: (Monad m) => Int -> Int -> MS.Stream m Int
stream !l !r = MS.Stream step l where { step x | x < r = return $ MS.Yield x (x + 1) | otherwise = return MS.Done; {-# INLINE [0] step #-}}
{-# INLINE [1] stream #-}
streamR :: (Monad m) => Int -> Int -> MS.Stream m Int
streamR !l !r = MS.Stream step (r - 1) where { step x | x >= l = return $ MS.Yield x (x - 1) | otherwise = return MS.Done; {-# INLINE [0] step #-}}
{-# INLINE [1] streamR #-}
stream' :: (Monad m) => Int -> Int -> Int -> MS.Stream m Int
stream' !l !r !d = MS.Stream step l where { step x | x < r = return $ MS.Yield x (x + d) | otherwise = return MS.Done; {-# INLINE [0] step #-}}
{-# INLINE [1] stream' #-}
infixl 8 `shiftRL`, `unsafeShiftRL`
shiftRL :: Int -> Int -> Int
shiftRL = unsafeShiftRL
{-# INLINE shiftRL #-}
unsafeShiftRL :: Int -> Int -> Int
unsafeShiftRL (I# x#) (I# i#) = I# (uncheckedIShiftRL# x# i#)
{-# INLINE unsafeShiftRL #-}
type Parser a = StateT C.ByteString Maybe a
runParser :: Parser a -> C.ByteString -> Maybe (a, C.ByteString)
runParser = runStateT
{-# INLINE runParser #-}
int :: Parser Int
int = coerce $ C.readInt . C.dropWhile isSpace
{-# INLINE int #-}
int1 :: Parser Int
int1 = fmap (subtract 1) int
{-# INLINE int1 #-}
char :: Parser Char
char = coerce C.uncons
{-# INLINE char #-}
byte :: Parser Word8
byte = coerce B.uncons
{-# INLINE byte #-}
skipSpaces :: Parser ()
skipSpaces = modify' (C.dropWhile isSpace)
{-# INLINE skipSpaces #-}
lowerBoundM :: (Monad m) => Int -> Int -> (Int -> m Bool) -> m Int
lowerBoundM low high p = go low high where { go !low !high | high <= low = return high | otherwise = p mid >>= bool (go (mid + 1) high) (go low mid) where { mid = low + unsafeShiftRL (high - low) 1}}
{-# INLINE lowerBoundM #-}
upperBoundM :: (Monad m) => Int -> Int -> (Int -> m Bool) -> m Int
upperBoundM low high p = do { flg <- p high; if flg then return high else subtract 1 <$!> lowerBoundM low high (fmap not . p)}
{-# INLINE upperBoundM #-}
lowerBound :: Int -> Int -> (Int -> Bool) -> Int
lowerBound low high p = runIdentity (lowerBoundM low high (return . p))
{-# INLINE lowerBound #-}
upperBound :: Int -> Int -> (Int -> Bool) -> Int
upperBound low high p = runIdentity (upperBoundM low high (return . p))
{-# INLINE upperBound #-}
-------------------------------------------------------------------------------
-- Data.IntMod
-------------------------------------------------------------------------------
modulus :: (Num a) => a
modulus = MOD
{-# INLINE modulus #-}
infixr 8 ^%
infixl 7 *%, /%
infixl 6 +%, -%
(+%) :: Int -> Int -> Int
(I# x#) +% (I# y#) = case x# +# y# of { r# -> I# (r# -# ((r# >=# MOD#) *# MOD#))}
{-# INLINE (+%) #-}
(-%) :: Int -> Int -> Int
(I# x#) -% (I# y#) = case x# -# y# of { r# -> I# (r# +# ((r# <# 0#) *# MOD#))}
{-# INLINE (-%) #-}
(*%) :: Int -> Int -> Int
(I# x#) *% (I# y#) = I# ((x# *# y#) `remInt#` MOD#)
{-# INLINE (*%) #-}
(/%) :: Int -> Int -> Int
(I# x#) /% (I# y#) = go# y# MOD# 1# 0# where { go# a# b# u# v# | isTrue# (b# ># 0#) = case a# `quotInt#` b# of { q# -> go# b# (a# -# (q# *# b#)) v# (u# -# (q# *# v#))} | otherwise = I# ((x# *# (u# +# MOD#)) `remInt#` MOD#)}
{-# INLINE (/%) #-}
(^%) :: Int -> Int -> Int
x ^% n | n > 0 = go 1 x n | n == 0 = 1 | otherwise = go 1 (1 /% x) (-n) where { go !acc !y !m | m .&. 1 == 0 = go acc (y *% y) (unsafeShiftR m 1) | m == 1 = acc *% y | otherwise = go (acc *% y) (y *% y) (unsafeShiftR (m - 1) 1)}
newtype IntMod = IntMod{getIntMod :: Int} deriving newtype (Eq, Ord, Read, Show, Real, Prim)
intMod :: (Integral a) => a -> IntMod
intMod x = fromIntegral $ mod (fromIntegral x) MOD
{-# INLINE intMod #-}
intModValidate :: IntMod -> Bool
intModValidate (IntMod x) = 0 <= x && x < MOD
{-# INLINE intModValidate #-}
instance Bounded IntMod where { minBound = IntMod 0; maxBound = IntMod $ modulus - 1}
instance Enum IntMod where { toEnum = intMod; fromEnum = coerce}
instance Integral IntMod where { quotRem x y = (x / y, x - x / y * y); toInteger = coerce (toInteger @Int)}
instance Num IntMod where { (+) = coerce (+%); (-) = coerce (-%); (*) = coerce (*%); abs = id; signum = const (IntMod 1); fromInteger x = coerce @Int @IntMod . fromInteger $ mod x modulus}
instance Fractional IntMod where { (/) = coerce (/%); fromRational q = fromInteger (numerator q) / fromInteger (denominator q)}
newtype instance  UM.MVector s IntMod = MV_IntMod (UM.MVector s Int)
newtype instance  U.Vector IntMod = V_IntMod (U.Vector Int)
instance U.Unbox IntMod
instance GM.MVector UM.MVector IntMod where { basicLength (MV_IntMod v) = GM.basicLength v; {-# INLINE basicLength #-}; basicUnsafeSlice i n (MV_IntMod v) = MV_IntMod $ GM.basicUnsafeSlice i n v; {-# INLINE basicUnsafeSlice #-}; basicOverlaps (MV_IntMod v1) (MV_IntMod v2) = GM.basicOverlaps v1 v2; {-# INLINE basicOverlaps #-}; basicUnsafeNew n = MV_IntMod `liftM` GM.basicUnsafeNew n; {-# INLINE basicUnsafeNew #-}; basicInitialize (MV_IntMod v) = GM.basicInitialize v; {-# INLINE basicInitialize #-}; basicUnsafeReplicate n x = MV_IntMod `liftM` GM.basicUnsafeReplicate n (coerce x); {-# INLINE basicUnsafeReplicate #-}; basicUnsafeRead (MV_IntMod v) i = coerce `liftM` GM.basicUnsafeRead v i; {-# INLINE basicUnsafeRead #-}; basicUnsafeWrite (MV_IntMod v) i x = GM.basicUnsafeWrite v i (coerce x); {-# INLINE basicUnsafeWrite #-}; basicClear (MV_IntMod v) = GM.basicClear v; {-# INLINE basicClear #-}; basicSet (MV_IntMod v) x = GM.basicSet v (coerce x); {-# INLINE basicSet #-}; basicUnsafeCopy (MV_IntMod v1) (MV_IntMod v2) = GM.basicUnsafeCopy v1 v2; {-# INLINE basicUnsafeCopy #-}; basicUnsafeMove (MV_IntMod v1) (MV_IntMod v2) = GM.basicUnsafeMove v1 v2; {-# INLINE basicUnsafeMove #-}; basicUnsafeGrow (MV_IntMod v) n = MV_IntMod `liftM` GM.basicUnsafeGrow v n; {-# INLINE basicUnsafeGrow #-}}
instance G.Vector U.Vector IntMod where { basicUnsafeFreeze (MV_IntMod v) = V_IntMod `liftM` G.basicUnsafeFreeze v; {-# INLINE basicUnsafeFreeze #-}; basicUnsafeThaw (V_IntMod v) = MV_IntMod `liftM` G.basicUnsafeThaw v; {-# INLINE basicUnsafeThaw #-}; basicLength (V_IntMod v) = G.basicLength v; {-# INLINE basicLength #-}; basicUnsafeSlice i n (V_IntMod v) = V_IntMod $ G.basicUnsafeSlice i n v; {-# INLINE basicUnsafeSlice #-}; basicUnsafeIndexM (V_IntMod v) i = coerce `liftM` G.basicUnsafeIndexM v i; {-# INLINE basicUnsafeIndexM #-}; basicUnsafeCopy (MV_IntMod mv) (V_IntMod v) = G.basicUnsafeCopy mv v; elemseq _ = seq; {-# INLINE elemseq #-}}