Problem:
Score : 400 points

Problem Statement

Takahashi will play a game using a piece on an array of squares numbered 1, 2, \cdots, N. Square i has an integer C_i written on it. Also, he is given a permutation of 1, 2, \cdots, N: P_1, P_2, \cdots, P_N.

Now, he will choose one square and place the piece on that square. Then, he will make the following move some number of times between 1 and K (inclusive):

In one move, if the piece is now on Square i (1 \leq i \leq N), move it to Square P_i. Here, his score increases by C_{P_i}.

Help him by finding the maximum possible score at the end of the game. (The score is 0 at the beginning of the game.)

Constraints

2 \leq N \leq 5000

1 \leq K \leq 10^9

1 \leq P_i \leq N

P_i \neq i

P_1, P_2, \cdots, P_N are all different.

-10^9 \leq C_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K
P_1 P_2 \cdots P_N
C_1 C_2 \cdots C_N

Output

Print the maximum possible score at the end of the game.

Sample Input 1

5 2
2 4 5 1 3
3 4 -10 -8 8

Sample Output 1

8

When we start at some square of our choice and make at most two moves, we have the following options:

If we start at Square 1, making one move sends the piece to Square 2, after which the score is 4. Making another move sends the piece to Square 4, after which the score is 4 + (-8) = -4.

If we start at Square 2, making one move sends the piece to Square 4, after which the score is -8. Making another move sends the piece to Square 1, after which the score is -8 + 3 = -5.

If we start at Square 3, making one move sends the piece to Square 5, after which the score is 8. Making another move sends the piece to Square 3, after which the score is 8 + (-10) = -2.

If we start at Square 4, making one move sends the piece to Square 1, after which the score is 3. Making another move sends the piece to Square 2, after which the score is 3 + 4 = 7.

If we start at Square 5, making one move sends the piece to Square 3, after which the score is -10. Making another move sends the piece to Square 5, after which the score is -10 + 8 = -2.

The maximum score achieved is 8.

Sample Input 2

2 3
2 1
10 -7

Sample Output 2

13

Sample Input 3

3 3
3 1 2
-1000 -2000 -3000

Sample Output 3

-1000

We have to make at least one move.

Sample Input 4

10 58
9 1 6 7 8 4 3 2 10 5
695279662 988782657 -119067776 382975538 -151885171 -177220596 -169777795 37619092 389386780 980092719

Sample Output 4

29507023469

The absolute value of the answer may be enormous.

Solution in Ruby (2.7.1) (judge verdict: Accepted)

class SegmentTree
  MIN = -Float::INFINITY
  def initialize(data)
    @offset = 1 << data.length.bit_length
    @data = Array.new(@offset * 2, MIN)
    data.each_with_index do |v, i|
      @data[@offset + i] = v
    end
    i = @offset - 1
    while i > 0
      @data[i] = merge(@data[2 * i], @data[2 * i + 1])
      i -= 1
    end
  end
  def merge(l, r); l > r ? l : r; end
  def max(a, b, k = 1, l = 0, r = @offset)
    mid = (l + r) / 2
    return MIN if b <= l || r <= a
    return @data[k] if a <= l && r <= b
    return merge(max(a,b,2 * k, l, mid), max(a,b,2 * k + 1, mid, r))
  end
end

def max(a,b); a > b ? a : b; end
def min(a,b); a < b ? a : b; end

def extract_cycle(s, used)
  cycle = []
  u = s
  until used[u]
    used[u] = true
    cycle << u  
    u = P[u]
  end
  return cycle
end

N, K = gets.split.map(&:to_i)
P = gets.split.map(&:to_i).tap{|array| array.unshift(0) }
C = gets.split.map(&:to_i).tap{|array| array.unshift(0) }

used = Array.new(N + 1)
ans = (1 .. N).inject(-1e20) do |max_score, s|
  next max_score if used[s]
  # sを始点とする閉路を抽出
  cycle = extract_cycle(s, used)
  z = cycle.size
  
  # 閉路の各頂点から最大1周分のスコアを計算できるように事前準備
  score = Array.new(3 * z + 1, 0)
  (1 .. z * 3).each do |i|
    v = cycle[i % z]
    score[i] = score[i - 1] + C[v]
  end

  # 閉路上の任意の点 s を始点としたときの最大スコアを 
  # セグ木を使って O(N log N) で計算
  
  sz = max(0, score[z])
  k = min(K, z + K % z)
  r = (K - k) / z

  st = SegmentTree.new(score)
  cycle_score = (0 ... z).map{|s| r * sz + st.max(s + 1, s + k + 1) - score[s]}.max

  max(max_score, cycle_score)
end

puts ans