output_description: Print the elements of s after Snuke finished the operations, in order from
  beginning to end, with spaces in between.
  
  * * *
input_description: Input is given from Standard Input in the following format:
  
      
      
      N K
      A_0 A_1 \cdots A_{N-1}
attempt: import sys
  
  readline = sys.stdin.readline
  MOD = 10**9 + 7
  INF = float("INF")
  sys.setrecursionlimit(10**5)
  
  
  def main():
      from collections import defaultdict
  
      N, K = map(int, readline().split())
      A = list(map(int, readline().split()))
  
      c = N * K
      b = 0
      while 2**b < c:
          b += 1
  
      doubling = [[0] * N for _ in range(b + 1)]
  
      idx = defaultdict(list)
      for i, x in enumerate(A):
          idx[x].append(i)
  
      for i in range(1, 200001):
          d = idx[i]
          l = len(d)
          for j in range(l):
              cur = d[j]
              nx = d[(j + 1) % l]
              if nx <= cur:
                  doubling[0][cur] = N - (cur - nx) + 1
              else:
                  doubling[0][cur] = nx - cur + 1
  
      for i in range(1, b + 1):
          for j in range(N):
              p = doubling[i - 1][j]
              doubling[i][j] = p + doubling[i - 1][(j + p) % N]
  
      cnt = 0
      cur = 0
      while c - cnt > 200000:
          for i in range(b, -1, -1):
              if cnt + doubling[i][cur] < c:
                  cnt += doubling[i][cur]
                  cur = cnt % N
                  break
  
      rem = c - cnt
      res = []
      res_set = set()
  
      while rem > 0:
          num = A[cur]
          if num in res_set:
              while True:
                  back = res[-1]
                  res.pop()
                  res_set.discard(back)
                  if back == num:
                      break
          else:
              res.append(num)
              res_set.add(num)
          rem -= 1
          cur += 1
          cur %= N
  
      print(*res)
  
  
  if __name__ == "__main__":
      main()
  
problem_description: Statement
  
  We have a sequence of N \times K integers: X=(X_0,X_1,\cdots,X_{N \times
  K-1}). Its elements are represented by another sequence of N integers:
  A=(A_0,A_1,\cdots,A_{N-1}). For each pair i, j (0 \leq i \leq K-1,\ 0 \leq j
  \leq N-1), X_{i \times N + j}=A_j holds.
  
  Snuke has an integer sequence s, which is initially empty. For each
  i=0,1,2,\cdots,N \times K-1, in this order, he will perform the following
  operation:
  
    * If s does not contain X_i: add X_i to the end of s.
    * If s does contain X_i: repeatedly delete the element at the end of s until s no longer contains X_i. Note that, in this case, we do **not** add X_i to the end of s.
  
  Find the elements of s after Snuke finished the operations.
samples: [{"input": "3 2\n    1 2 3", "output": "2 3\n    \n\nIn this case, X=(1,2,3,1,2,3). We will perform the operations as follows:\n\n  * i=0: s does not contain 1, so we add 1 to the end of s, resulting in s=(1).\n  * i=1: s does not contain 2, so we add 2 to the end of s, resulting in s=(1,2).\n  * i=2: s does not contain 3, so we add 3 to the end of s, resulting in s=(1,2,3).\n  * i=3: s does contain 1, so we repeatedly delete the element at the end of s as long as s contains 1, which causes the following changes to s: (1,2,3)\u2192(1,2)\u2192(1)\u2192().\n  * i=4: s does not contain 2, so we add 2 to the end of s, resulting in s=(2).\n  * i=5: s does not contain 3, so we add 3 to the end of s, resulting in s=(2,3).\n\n* * *"}, {"input": "5 10\n    1 2 3 2 3", "output": "3\n    \n\n* * *"}, {"input": "6 1000000000000\n    1 1 2 2 3 3", "output": "s may be empty in the end.\n\n* * *"}, {"input": "11 97\n    3 1 4 1 5 9 2 6 5 3 5", "output": "9 2 6"}]