Problem:
Score : 100 points

Problem Statement

There are N rabbits, numbered 1, 2, \ldots, N.

For each i, j (1 \leq i, j \leq N), the compatibility of Rabbit i and j is described by an integer a_{i, j}.
Here, a_{i, i} = 0 for each i (1 \leq i \leq N), and a_{i, j} = a_{j, i} for each i and j (1 \leq i, j \leq N).

Taro is dividing the N rabbits into some number of groups.
Here, each rabbit must belong to exactly one group.
After grouping, for each i and j (1 \leq i < j \leq N), Taro earns a_{i, j} points if Rabbit i and j belong to the same group.

Find Taro's maximum possible total score.

Constraints

All values in input are integers.

1 \leq N \leq 16

|a_{i, j}| \leq 10^9

a_{i, i} = 0

a_{i, j} = a_{j, i}

Input

Input is given from Standard Input in the following format:

N
a_{1, 1} \ldots a_{1, N}
:
a_{N, 1} \ldots a_{N, N}

Output

Print Taro's maximum possible total score.

Sample Input 1

3
0 10 20
10 0 -100
20 -100 0

Sample Output 1

20

The rabbits should be divided as \{1, 3\}, \{2\}.

Sample Input 2

2
0 -10
-10 0

Sample Output 2

0

The rabbits should be divided as \{1\}, \{2\}.

Sample Input 3

4
0 1000000000 1000000000 1000000000
1000000000 0 1000000000 1000000000
1000000000 1000000000 0 -1
1000000000 1000000000 -1 0

Sample Output 3

4999999999

The rabbits should be divided as \{1, 2, 3, 4\}.
Note that the answer may not fit into a 32-bit integer type.

Sample Input 4

16
0 5 -4 -5 -8 -4 7 2 -4 0 7 0 2 -3 7 7
5 0 8 -9 3 5 2 -7 2 -7 0 -1 -4 1 -1 9
-4 8 0 -9 8 9 3 1 4 9 6 6 -6 1 8 9
-5 -9 -9 0 -7 6 4 -1 9 -3 -5 0 1 2 -4 1
-8 3 8 -7 0 -5 -9 9 1 -9 -6 -3 -8 3 4 3
-4 5 9 6 -5 0 -6 1 -2 2 0 -5 -2 3 1 2
7 2 3 4 -9 -6 0 -2 -2 -9 -3 9 -2 9 2 -5
2 -7 1 -1 9 1 -2 0 -6 0 -6 6 4 -1 -7 8
-4 2 4 9 1 -2 -2 -6 0 8 -6 -2 -4 8 7 7
0 -7 9 -3 -9 2 -9 0 8 0 0 1 -3 3 -6 -6
7 0 6 -5 -6 0 -3 -6 -6 0 0 5 7 -1 -5 3
0 -1 6 0 -3 -5 9 6 -2 1 5 0 -2 7 -8 0
2 -4 -6 1 -8 -2 -2 4 -4 -3 7 -2 0 -9 7 1
-3 1 1 2 3 3 9 -1 8 3 -1 7 -9 0 -6 -8
7 -1 8 -4 4 1 2 -7 7 -6 -5 -8 7 -6 0 -9
7 9 9 1 3 2 -5 8 7 -6 3 0 1 -8 -9 0

Sample Output 4

132

Solution in Go (1.6) (judge verdict: Time Limit Exceeded)

package main

import (
	"bufio"
	"fmt"
	"os"
	"strconv"
	"strings"
)

func sum(a []int) int {
	total := 0
	for _, x := range a {
		total = (total + x) % CYCLE
	}
	return total
}

func max(a int64, b int64) int64 {
	if a > b {
		return a
	} else {
		return b
	}
}

func parseInt(s string) int {
	return int(parseInt64(s))
}

func parseInt64(s string) int64 {
	x, _ := strconv.ParseInt(s, 10, 64)
	return x
}

var reader *bufio.Reader

func readLine() string {
	s, _ := reader.ReadString('\n')
	return strings.TrimSuffix(s, "\n")
}

var CYCLE = 1000000007

var a [][]int64
var N int

func printTable(t [][]int64) {
	for i := 0; i < len(t); i++ {
		fmt.Print(t[i])
		fmt.Print(" ")
	}
	fmt.Println("")
}

func reduce(a [][]int64, p, q int) [][]int64 {
	n := len(a)
	b := make([][]int64, n-1)
	for i := 0; i < n; i++ {
		var ii int
		if i == q {
			ii = p
		} else {
			if i < q {
				ii = i
			} else {
				ii = i - 1
			}
			b[ii] = make([]int64, n-1)
		}
		for j := 0; j < n; j++ {
			jj := j
			if j == q {
				jj = p
			} else if j > q {
				jj--
			}
			b[ii][jj] += a[i][j]
		}
	}
	b[p][p] -= a[p][q]
	return b
}

func diagTotal(a [][]int64) int64 {
	var result int64
	for i := 0; i < len(a); i++ {
		result += a[i][i]
	}
	return result
}

func reduceRecursively(a [][]int64) int64 {
	// printTable(a)
	n := len(a)
	result := diagTotal(a)
	for i := 0; i < n-1; i++ {
		for j := i + 1; j < n; j++ {
			r := reduceRecursively(reduce(a, i, j))
			if r > result {
				result = r
			}
		}
	}
	return result
}

func main() {
	reader = bufio.NewReader(os.Stdin)
	N = parseInt(readLine())
	a = make([][]int64, N)
	for i := 0; i < N; i++ {
		a[i] = make([]int64, N)
		line := readLine()
		for j, token := range strings.Fields(line) {
			a[i][j] = parseInt64(token)
		}
	}

	fmt.Println(reduceRecursively(a))
}