Problem:
Score : 400 points

Problem Statement

We have an H-by-W matrix.
Let a_{ij} be the element at the i-th row from the top and j-th column from the left.
In this matrix, each a_{ij} is a lowercase English letter.

Snuke is creating another H-by-W matrix, A', by freely rearranging the elements in A.
Here, he wants to satisfy the following condition:

Every row and column in A' can be read as a palindrome.

Determine whether he can create a matrix satisfying the condition.

Note

A palindrome is a string that reads the same forward and backward.
For example, a, aa, abba and abcba are all palindromes, while ab, abab and abcda are not.

Constraints

1 ≤ H, W ≤ 100

a_{ij} is a lowercase English letter.

Input

Input is given from Standard Input in the following format:

H W
a_{11}a_{12}...a_{1W}
:
a_{H1}a_{H2}...a_{HW}

Output

If Snuke can create a matrix satisfying the condition, print Yes; otherwise, print No.

Sample Input 1

3 4
aabb
aabb
aacc

Sample Output 1

Yes

For example, the following matrix satisfies the condition.

abba
acca
abba

Sample Input 2

2 2
aa
bb

Sample Output 2

No

It is not possible to create a matrix satisfying the condition, no matter how we rearrange the elements in A.

Sample Input 3

5 1
t
w
e
e
t

Sample Output 3

Yes

For example, the following matrix satisfies the condition.

t
e
w
e
t

Sample Input 4

2 5
abxba
abyba

Sample Output 4

No

Sample Input 5

1 1
z

Sample Output 5

Yes

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

import qualified Data.Map.Strict as Map

main = do
    [rows, columns] <- fmap (fmap read . words) getLine
    allLetters <- buildMap . concat <$> mapM (const getLine) [1..rows]
    let numFours = sum . fmap (`div` 4) $ allLetters
    let numPairs = sum . fmap (`div` 2) $ allLetters
    let neededFours = if columns > 1 && rows > 1 then (max rows columns) `div` 2 else 0
    let neededColumnPairs = if odd rows then columns `div` 2 else 0
    let neededRowPairs = if odd columns then rows `div` 2 else 0
    let remainingPairs = numPairs - neededFours * 2
    if numFours >= neededFours && remainingPairs >= (neededColumnPairs + neededRowPairs) 
        then putStrLn "Yes" 
        else putStrLn "No"
    
buildMap = foldl (\m c -> Map.insertWith (+) c 1 m) Map.empty