Problem:
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Dice IV

Write a program which reads $n$ dices constructed in the same way as Dice I, and determines whether they are all different. For the determination, use the same way as Dice III.

Input

In the first line, the number of dices $n$ is given. In the following $n$ lines, six integers assigned to the dice faces are given respectively in the same way as Dice III.

Output

Print "Yes" if given dices are all different, otherwise "No" in a line.

Constraints

$2 \leq n \leq 100$

$0 \leq $ the integer assigned to a face $ \leq 100$

Sample Input 1

3
1 2 3 4 5 6
6 2 4 3 5 1
6 5 4 3 2 1

Sample Output 1

No

Sample Input 2

3
1 2 3 4 5 6
6 5 4 3 2 1
5 4 3 2 1 6

Sample Output 2

Yes

Solution in Ruby (judge verdict: Runtime Error)

class DiceError < StandardError
end

n = gets.to_i
dices = []
n.times {dices << gets.split.map(&:to_i)}

dices2 = dices.map(&:sort).uniq
group = Array.new(dices2.size)
dices2.each_with_index do |d2, i|
  group[i] = dices.select {|d| d.sort == d2}
end

throw = {E: [3, 2, 6, 1, 5, 4], N: [5, 1, 3, 4, 6, 2],
         R: [1, 4, 2, 5, 3, 6]}
th = {}
throw.each_key {|k| th[k] = throw[k].map(&:pred)}

rotate = ->(dir, dice) {
  next_dice = Array.new(6, 0)
  dice.each_with_index do |d, i|
    next_dice[th[dir][i]] = d
  end
  next_dice
}

adapt_same_side = ->(dice) {
  stack = [:N, :N, :N, :N, :E, :N, :N, :N, :N]
  stack.each do |dir|
    if dice[0] == dices[0][0]
      return dice
    else
      dice = rotate.(dir, dice)
    end
  end
  raise DiceError
}

is_same_dices = ->(dice1, dice2) {
  result = false
  4.times do
    if dice2 == dice1
      result = true
      break
    end
    dice2 = rotate.(:R, dice2)
  end
  result
}

handle_group = ->(dices) {
  n = dices.size
  new_dices = [dices[0]]
  dices[1, n - 1].each {|d| new_dices << adapt_same_side.(d)}
  new_dices.combination(2) do |dice1, dice2|
    return false if is_same_dices.(dice1, dice2)
  end
  true
}

result = "Yes"
group.each do |dices|
  result = "No" unless handle_group.(dices)
end
puts result