Problem:
Score : 500 points

Problem Statement

N players will participate in a tennis tournament. We will call them Player 1, Player 2, \ldots, Player N.

The tournament is round-robin format, and there will be N(N-1)/2 matches in total.
Is it possible to schedule these matches so that all of the following conditions are satisfied? If the answer is yes, also find the minimum number of days required.

Each player plays at most one matches in a day.

Each player i (1 \leq i \leq N) plays one match against Player A_{i, 1}, A_{i, 2}, \ldots, A_{i, N-1} in this order.

Constraints

3 \leq N \leq 1000

1 \leq A_{i, j} \leq N

A_{i, j} \neq i

A_{i, 1}, A_{i, 2}, \ldots, A_{i, N-1} are all different.

Input

Input is given from Standard Input in the following format:

N
A_{1, 1} A_{1, 2} \ldots A_{1, N-1}
A_{2, 1} A_{2, 2} \ldots A_{2, N-1}
:
A_{N, 1} A_{N, 2} \ldots A_{N, N-1}

Output

If it is possible to schedule all the matches so that all of the conditions are satisfied, print the minimum number of days required; if it is impossible, print -1.

Sample Input 1

3
2 3
1 3
1 2

Sample Output 1

3

All the conditions can be satisfied if the matches are scheduled for three days as follows:

Day 1: Player 1 vs Player 2

Day 2: Player 1 vs Player 3

Day 3: Player 2 vs Player 3

This is the minimum number of days required.

Sample Input 2

4
2 3 4
1 3 4
4 1 2
3 1 2

Sample Output 2

4

All the conditions can be satisfied if the matches are scheduled for four days as follows:

Day 1: Player 1 vs Player 2, Player 3 vs Player 4

Day 2: Player 1 vs Player 3

Day 3: Player 1 vs Player 4, Player 2 vs Player 3

Day 4: Player 2 vs Player 4

This is the minimum number of days required.

Sample Input 3

3
2 3
3 1
1 2

Sample Output 3

-1

Any scheduling of the matches violates some condition.

Solution in LuaJIT (2.0.4) (judge verdict: Time Limit Exceeded)

local n = io.read("*n")
local t = {}
local pos = {}
local waitlist = {}
for i = 1, n do
  pos[i] = 1
  t[i] = {}
  for j = 1, n - 1 do
    t[i][j] = io.read("*n")
  end
end

for i = 1, n do
  if not waitlist[t[i][1]] then
    waitlist[t[i][1]] = {}
  end
  waitlist[t[i][1]][i] = true
end

local ret = 0
while true do
  ret = ret + 1
  local proceed = false
  local remain = false
  local done = {}
  for i = 1, n do done[i] = false end
  for i, dstlist in pairs(waitlist) do
    if pos[i] < n and not done[i] then
      for nxt, _unused in pairs(dstlist) do
        remain = true
        if waitlist[nxt] and waitlist[nxt][i] and not done[nxt] then
          proceed = true
          waitlist[nxt][i] = nil
          waitlist[i][nxt] = nil
          pos[i] = pos[i] + 1
          pos[nxt] = pos[nxt] + 1
          done[i] = true
          done[nxt] = true
          if pos[i] < n then
            if not waitlist[t[i][pos[i]]] then waitlist[t[i][pos[i]]] = {} end
            waitlist[t[i][pos[i]]][i] = true
          end
          if pos[nxt] < n then
            if not waitlist[t[nxt][pos[nxt]]] then waitlist[t[nxt][pos[nxt]]] = {} end
            waitlist[t[nxt][pos[nxt]]][nxt] = true
          end
          break
        end
      end
    end
  end
  if not remain then
    if not proceed then
      ret = ret - 1
    end
    break
  end
  if not proceed then
    ret = -1
    break
  end
end
print(ret)