Problem:
Score : 600 points

Problem Statement

We have 3N cards arranged in a row from left to right, where each card has an integer between 1 and N (inclusive) written on it. The integer written on the i-th card from the left is A_i.

You will do the following operation N-1 times:

Rearrange the five leftmost cards in any order you like, then remove the three leftmost cards. If the integers written on those three cards are all equal, you gain 1 point.

After these N-1 operations, if the integers written on the remaining three cards are all equal, you will gain 1 additional point.

Find the maximum number of points you can gain.

Constraints

1 \leq N \leq 2000

1 \leq A_i \leq N

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 \cdots A_{3N}

Output

Print the maximum number of points you can gain.

Sample Input 1

2
1 2 1 2 2 1

Sample Output 1

2

Let us rearrange the five leftmost cards so that the integers written on the six cards will be 2\ 2\ 2\ 1\ 1\ 1 from left to right.

Then, remove the three leftmost cards, all of which have the same integer 2, gaining 1 point.

Now, the integers written on the remaining cards are 1\ 1\ 1.

Since these three cards have the same integer 1, we gain 1 more point.

In this way, we can gain 2 points - which is the maximum possible.

Sample Input 2

3
1 1 2 2 3 3 3 2 1

Sample Output 2

1

Sample Input 3

3
1 1 2 2 2 3 3 3 1

Sample Output 3

3

Solution in Ruby (2.7.1) (judge verdict: Wrong Answer)

INIT = -10_000_000
N = gets.to_i
S = gets.split.map(&:to_i)

dp1 = Array.new(N + 1, INIT)
class << dp1
  def chmax(i, v)
    self[i] = v if v > self[i]
  end
end

dp2 = Array.new(N + 1){ Array.new(N + 1, INIT) }
class << dp2
  def chmax(i,j,v)
    self[i][j] = v if v > self[i][j]
  end
end

S[0,2].sort.tap do |a,b|
  dp2.chmax(a,b,0)
  dp1.chmax(a,0)
  dp1.chmax(b,0)
end

q = []
base = 0

(1 .. N - 1).each do |i|
  c,d,e = S[3 * i - 1, 3].sort
  if c == e
    next base += 1
  elsif c == d
    (1 .. N).each do |x|
      if x < c
        q << [ x, e, dp2[x][c] + 1] if dp2[x][c] >= 0 && dp2[x][e] < dp2[x][c] + 1
      elsif x < e
        q << [ x, e, dp2[c][x] + 1] if dp2[c][x] >= 0 && dp2[x][e] < dp2[c][x] + 1
      else # _ >= e
        q << [ e, x, dp2[e][x] + 1] if dp2[e][x] >= 0 && dp2[e][x] < dp2[e][x] + 1
      end
    end
  elsif d == e
    (1 .. N).each do |x|
      if x < c
        q << [ x, c, dp2[x][e] + 1] if dp2[x][e] >= 0 && dp2[x][c] < dp2[x][e] + 1
      elsif x < e
        q << [ c, x, dp2[x][e] + 1] if dp2[x][c] >= 0 && dp2[c][x] < dp2[x][e] + 1
      else
        q << [ c, x, dp2[e][x] + 1] if dp2[e][x] >= 0 && dp2[c][x] < dp2[e][x] + 1
      end
    end
  else
    q << [ d, e, dp2[c][c] + 1] if dp2[c][c] >= 0 && dp2[d][e] < dp2[c][c] + 1
    q << [ c, e, dp2[d][d] + 1] if dp2[d][d] >= 0 && dp2[c][e] < dp2[d][d] + 1
    q << [ c, d, dp2[e][e] + 1] if dp2[e][e] >= 0 && dp2[c][d] < dp2[e][e] + 1
  end

  (1 .. N).each do |x|
    next if dp1[x] < 0
    [c,d,e].each do |y|
      if x < y
        q << [ x, y, dp1[x]] if dp2[x][y] < dp1[x]
      else
        q << [ y, x, dp1[x]] if dp2[y][x] < dp1[x]
      end
    end
  end

  until q.empty?
    a,b,v = q.shift
    if dp2.chmax(a,b,v)
      dp1.chmax(a,v)
      dp1.chmax(b,v)
    end
  end
end

s = S[-1]
puts [base + dp2[s][s] + 1, base + dp1.max].max