Problem:


Score : 700 points

Problem 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.

Constraints

1 \leq N \leq 2 \times 10^5

1 \leq K \leq 10^{12}

1 \leq A_i \leq 2 \times 10^5

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K
A_0 A_1 \cdots A_{N-1}

Output

Print the elements of s after Snuke finished the operations, in order from beginning to end, with spaces in between.

Sample Input 1

3 2
1 2 3

Sample Output 1

2 3

In this case, X=(1,2,3,1,2,3).
We will perform the operations as follows:

i=0: s does not contain 1, so we add 1 to the end of s, resulting in s=(1).

i=1: s does not contain 2, so we add 2 to the end of s, resulting in s=(1,2).

i=2: s does not contain 3, so we add 3 to the end of s, resulting in s=(1,2,3).

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)→(1,2)→(1)→().

i=4: s does not contain 2, so we add 2 to the end of s, resulting in s=(2).

i=5: s does not contain 3, so we add 3 to the end of s, resulting in s=(2,3).

Sample Input 2

5 10
1 2 3 2 3

Sample Output 2

3

Sample Input 3

6 1000000000000
1 1 2 2 3 3

Sample Output 3

s may be empty in the end.

Sample Input 4

11 97
3 1 4 1 5 9 2 6 5 3 5

Sample Output 4

9 2 6



Sample input:
3 2
1 2 3


Sample output:
2 3

Submission status: Runtime Error

Solution code:
N,K = map(int,input().split())
A = list(map(int,input().split()))

AA = A+A
last_i = {}
nx = [-1]*N
for i,a in enumerate(reversed(AA)):
    i = 2*N-i-1
    if i < N:
        nx[i] = (last_i[a] - i) % N + 1
    last_i[a] = i

visited = [-1]*N
visited[0] = 0
j = i = 0
while 1:
    p = nx[i]+N if nx[i]==1 else nx[i]
    j += p
    if visited[j%N] < 0:
        visited[j%N] = j
    else:
        break
    i = j%N
cycle = (j - visited[j%N]) // N

K %= cycle
j = i = 0
while 1:
    p = nx[i]+N if nx[i]==1 else nx[i]
    if j+p > N*K:
        break
    j += p
    i = j%N

if j%N == 0:
    print()
else:
    ans = []
    anss = set()
    for a in A[j%N:]:
        if a in anss:
            anss.remove(a)
            while ans[-1] != a:
                ans.pop()
            ans.pop()
        else:
            anss.add(a)
            ans.append(a)
    print(*ans)

Submission status: Runtime Error

Solution code:
N,K = map(int,input().split())
A = list(map(int,input().split()))

AA = A+A
last_i = {}
dp = [[0] for i in range(N)]
for i,a in enumerate(reversed(AA)):
    i = 2*N-i-1
    if i < N:
        dp[i][0] = last_i[a] - i + 1
    last_i[a] = i

MAX = N*K
k = 0
while (1<<k) <= MAX:
    for i in range(N):
        d,m = divmod(dp[i][k]+i, N)
        v = dp[m][k]
        dp[i].append(v + d*N+m-i)
    k += 1

def jump(k):
    j = x = 0
    while (1<<x) <= k:
        if k&(1<<x):
            j += dp[j%N][x]
            if j > MAX:
                return MAX+1
        x += 1
    return j

ok = 0
ng = MAX
while ng-ok > 1:
    m = (ok+ng)//2
    j = jump(m)
    if j <= MAX:
        ok = m
    else:
        ng = m

s = jump(ok)%N
if s==0:
    print()
else:
    ans = []
    anss = set()
    for a in A[s:]:
        if a in anss:
            anss.remove(a)
            while ans[-1] != a:
                ans.pop()
            ans.pop()
        else:
            anss.add(a)
            ans.append(a)
    print(*ans)

Submission status: Runtime Error

Solution code:
N,K = map(int,input().split())
A = list(map(int,input().split()))

AA = A+A
last_i = {}
dp = [[0] for i in range(N)]
for i,a in enumerate(reversed(AA)):
    i = 2*N-i-1
    if i < N:
        dp[i][0] = last_i[a] - i + 1
    last_i[a] = i

MAX = N*K
k = 0
while (1<<k) <= MAX:
    for i in range(N):
        d,m = divmod(dp[i][k]+i, N)
        v = dp[m][k]
        dp[i].append(v + d*N+m-i)
    k += 1

def jump(k):
    j = x = 0
    while (1<<x) <= k:
        if k&(1<<x):
            j += dp[j%N][x]
            if j > MAX:
                return MAX+1
        x += 1
    return j

ok = 0
ng = MAX
jp = 0
while ng-ok > 1:
    m = (ok+ng)//2
    j = jump(m)
    if j <= MAX:
        ok = m
        jp = j
    else:
        ng = m
s = jp%N
if s==0:
    print()
else:
    ans = []
    anss = set()
    for a in A[s:]:
        if a in anss:
            anss.remove(a)
            while ans[-1] != a:
                ans.pop()
            ans.pop()
        else:
            anss.add(a)
            ans.append(a)
    print(*ans)

Submission status: Runtime Error

Solution code:
import itertools
from bisect import bisect_left,bisect
import sys

n,K=map(int,input().split())
A=list(map(int,input().split()))

A_max=max(A)
lst=[[] for i in range(A_max+1)]
flag=[0]*(A_max+1)
for i in range(n):
    lst[A[i]].append(i)

flag=[0]*n
data=[]
i=0
while flag[i]==0:
    flag[i]=1
    a=A[i]
    b=bisect(lst[a],i)
    if b==len(lst[a]):
        data.append(i)
        i=lst[a][0]+1
    else:
        i=lst[a][b]+1
    if i==n:
        data.append(n)
        i=0

#大丈夫
if not data:
    print()
    sys.exit()
s=len(data)
zzz=data[(K-1)%s]
i=zzz
d=[]
while i<n:
    a=A[i]
    b=bisect(lst[a],i)
    if b<len(lst[a]):
        d.append([i,lst[a][b]])
        i=lst[a][b]+1
        continue
    else:
        i+=1


if d:
    ans=A[zzz:d[0][0]]
    for j in range(len(d)-1):
        ans=list(itertools.chain(ans,A[d[i][1]+1:d[i+1][0]]))
    ans=list(itertools.chain(ans,A[d[-1][1]+1:]))
    print(*ans)

Submission status: Runtime Error

Solution code:
import itertools
from bisect import bisect_left,bisect
import sys

n,K=map(int,input().split())
A=list(map(int,input().split()))

A_max=max(A)
lst=[[] for i in range(A_max+1)]
flag=[0]*(A_max+1)
for i in range(n):
    lst[A[i]].append(i)

flag=[0]*n
data=[]
i=0
while flag[i]==0:
    flag[i]=1
    a=A[i]
    b=bisect(lst[a],i)
    if b==len(lst[a]):
        data.append(i)
        i=lst[a][0]+1
    else:
        i=lst[a][b]+1
    if i==n:
        data.append(n)
        i=0

#大丈夫
if not data:
    print()
    sys.exit()
s=len(data)
zzz=data[(K-1)%s]
i=zzz
d=[]
while i<n:
    a=A[i]
    b=bisect(lst[a],i)
    if b<len(lst[a]):
        d.append([i,lst[a][b]])
        i=lst[a][b]+1
        continue
    else:
        i+=1


if d:
    ans=A[zzz:d[0][0]]
    for j in range(len(d)-1):
        ans=list(itertools.chain(ans,A[d[i][1]+1:d[i+1][0]]))
    ans=list(itertools.chain(ans,A[d[-1][1]+1:]))
    print(*ans)
else:
    print(*A[zzz:])

Submission status: Runtime Error

Solution code:
import itertools
from bisect import bisect_left,bisect
import sys

n,K=map(int,input().split())
A=list(map(int,input().split()))

A_max=max(A)
lst=[[] for i in range(A_max+1)]
flag=[0]*(A_max+1)
for i in range(n):
    lst[A[i]].append(i)

flag=[0]*n
data=[]
i=0
while flag[i]==0:
    flag[i]=1
    a=A[i]
    b=bisect(lst[a],i)
    if b==len(lst[a]):
        data.append(i)
        i=lst[a][0]+1
    else:
        i=lst[a][b]+1
    if i==n:
        data.append(n)
        i=0

if not data:
    print()
    sys.exit()
s=len(data)
zzz=data[(K-1)%s]
i=zzz
d=[]
while i<n:
    a=A[i]
    b=bisect(lst[a],i)
    if b<len(lst[a]):
        d.append([i,lst[a][b]])
        i=lst[a][b]+1
        continue
    else:
        i+=1

if d:
    ans=A[zzz:d[0][0]]
    for j in range(len(d)-1):
        ans=itertools.chain(ans,A[d[i][1]+1:d[i+1][0]])
    ans=itertools.chain(ans,A[d[-1][1]+1:])
    print(*ans)
else:
    print(*A[zzz:])

Submission status: Runtime Error

Solution code:
import itertools
from bisect import bisect_left,bisect

n,K=map(int,input().split())
A=list(map(int,input().split()))

A_max=max(A)
lst=[[] for i in range(A_max+1)]
flag=[0]*(A_max+1)
for i in range(n):
    lst[A[i]].append(i)

flag=[0]*n
data=[]
i=0
while flag[i]==0:
    flag[i]=1
    a=A[i]
    b=bisect(lst[a],i)
    if b==len(lst[a]):
        data.append(i)
        i=lst[a][0]+1
    else:
        i=lst[a][b]+1
    if i==n:
        data.append(n)
        i=0

s=len(data)
zzz=data[(K-1)%s]
i=zzz
d=[]
while i<n:
    a=A[i]
    if lst[a]:
        b=bisect(lst[a],i)
        if b<len(lst[a]):
            d.append([i,lst[a][b]])
            i=lst[a][b]+1
            continue
    i+=1

if d:
    ans=A[zzz:d[0][0]]
    for j in range(len(d)-1):
        ans=itertools.chain(ans,A[d[i][1]+1:d[i+1][0]])
    ans=itertools.chain(ans,A[d[-1][1]+1:])
    print(*ans)
else:
    print(*A[zzz:])

Submission status: Runtime Error

Solution code:
import sys
sys.setrecursionlimit(2147483647)
INF=float("inf")
MOD=10**9+7
input=lambda:sys.stdin.readline().rstrip()
def resolve():
    n,k=map(int,input().split())
    A=list(map(int,input().split()))
    idx=[[] for _ in range(max(A)+1)]
    for i,a in enumerate(A):
        idx[a].append(i)
    perm=[None]*n
    for I in idx:
        s=len(I)
        for i in range(s):
            if(i!=s-1): perm[I[i]]=(I[i+1]+1)%n
            else: perm[I[i]]=(I[0]+1)%n

    # perm の 先頭要素をサイクル分解する
    cycle=[0]
    v=perm[0]
    while(v):
        cycle.append(v)
        v=perm[v]

    # 元に戻るまで何周したか -> まずは k をこれで割る
    cycle.append(0)
    cnt=0
    for i in range(len(cycle)-1):
        if(cycle[i+1]<cycle[i] or cycle[i]+1==cycle[i+1]):
            cnt+=1
    k%=cnt

    # k が 0 になるまで消費する
    for i in range(len(cycle)-1):
        s=(cycle[i+1]<cycle[i] or cycle[i]+1==cycle[i+1])
        if(k-s<=0):
            break
        else:
            k-=s

    # cycle[i] 以降の数列について、実際に構成すればよい
    Q=[]
    cnt=[0]*(max(A)+1)
    for a in A[cycle[i]:]:
        if(cnt[a]==0):
            Q.append(a)
            cnt[a]=1
        else:
            while(Q[-1]!=a): Q.pop()
            Q.pop()
            cnt[a]=0
    print(*Q)
resolve()

Submission status: Runtime Error

Solution code:
import sys
sys.setrecursionlimit(2147483647)
INF=float("inf")
MOD=10**9+7
input=lambda:sys.stdin.readline().rstrip()
def resolve():
    n,k=map(int,input().split())
    A=list(map(int,input().split()))
    idx=[[] for _ in range(max(A)+1)]
    for i,a in enumerate(A):
        idx[a].append(i)
    perm=[None]*n
    for I in idx:
        s=len(I)
        for i in range(s):
            if(i!=s-1): perm[I[i]]=(I[i+1]+1)%n
            else: perm[I[i]]=(I[0]+1)%n

    # perm の 先頭要素をサイクル分解する
    cycle=[0]
    v=perm[0]
    while(v):
        cycle.append(v)
        v=perm[v]

    # 元に戻るまで何周したか -> まずは k をこれで割る
    cycle.append(0)
    cnt=0
    for i in range(len(cycle)-1):
        if(cycle[i+1]<cycle[i] or cycle[i]+1==cycle[i+1]):
            cnt+=1
    k%=cnt

    # k が 0 になるまで消費する
    for i in range(len(cycle)-1):
        s=(cycle[i+1]<cycle[i] or cycle[i]+1==cycle[i+1])
        if(k-s<=0):
            break
        else:
            k-=s

    # cycle[i] 以降の数列について、実際に構成すればよい
    ans=[]
    i=cycle[i]
    while(i!=0 and i!=n):
        if(i==n-1):
            ans.append(A[i])
            i+=1
        elif(perm[i]==0):
            i=n
        elif(perm[i]==i+1 or perm[i]<i):
            ans.append(A[i])
            i+=1
        else:
            i=perm[i]
    print(*ans)
resolve()

Submission status: Runtime Error

Solution code:
def main():
    import bisect
    n,k=map(int,input().split())
    x=list(map(int,input().split()))
    d=dict()
    for j,i in enumerate(x):
        if i in d.keys():
            d[i].append(j)
        else:
            d[i]=[j]
    first_word=[""]
    
    def operation(temp_operated,temp_operator):
        l=[i for i in temp_operated]
        s=set(temp_operated)
        for i in temp_operator:
            if i in s:
                while True:
                    if l[-1]!=i:
                        l.pop()
                    else:
                        l.pop()
                        s=set(l)
                        break
            else:
                l.append(i)
                s.add(i)
        return l
    
    if operation("",x)!=[]:
        first_word.append(operation("",x)[0])
    flag=0

    while True:
        w=first_word[-1]
        if w=="":
            break
        t=d[w][0]
        if t==n-1:
            break
        while True:
            ww=x[t+1]
            temp=bisect.bisect_right(d[ww],t+1)
            if temp>=len(d[ww]):
                first_word.append(ww)
                break
            else:
                t=d[ww][temp]
                if t==n-1:
                    flag=1
                    break
        if flag==1:
            break
    
    l=len(first_word)
    k=k%l
    if len(first_word)==1:
        print("")
    else:
        print(*operation("",x[d[first_word[k-1]][0]+1:]))

main()

Submission status: Runtime Error

Solution code:
n,k,*a=map(int,open(0).read().split())
l,r,*b=[[]for _ in range(10**6)]
for i,v in enumerate(a):
  b[v]+=i,
  l+=len(b[v]),
s=a[0]
i=c=0
while 1:
  j=l[i]%len(b[s])
  if(c>k-2)>j:
    r+=s,
    i+=1
  else:i=-~b[s][j]%n
  c+=j<1
  if i<1:
    if c>k-2:break
    k%=c+1
    c=0
  s=a[i]
print(*r)

Submission status: Runtime Error

Solution code:
from bisect import*
from collections import*
n,k,*a=map(int,open(0).read().split())
b=defaultdict(list)
for i,v in enumerate(a):b[v]+=i,
s=a[0]
i=c=0
while 1:
  t=b[s]
  m=len(t)
  j=bisect(t,i)
  c+=j==m
  i=t[j%m]+1
  if i==n:break
  s=a[i]
k%=c+1
"""
s=[]
for i in a*k:
  if i in s:s=s[:s.index(i)]
  else:s+=i,
print(*s)
"""
a*k

Submission status: Runtime Error

Solution code:
from bisect import*
n,k,*a=map(int,open(0).read().split())
b=[[]for _ in range(max(a)+1)]
for i,v in enumerate(a):b[v]+=i,
s=a[0]
i=c=0
while 1:
  t=b[s]
  m=len(t)
  j=bisect(t,i)
  c+=j==m
  i=t[j%m]+1
  if i==n:break
  s=a[i]
k%=c+1
s=[]
for i in a*k:
  if i in s:s=s[:s.index(i)]
  else:s+=i,
print(*s)

Submission status: Runtime Error

Solution code:
from bisect import*
n,k,*a=map(int,open(0).read().split())
b=[[]for _ in range(n+1)]
for i,v in enumerate(a):b[v]+=i,
s=a[0]
i=c=0
while 1:
  t=b[s]
  m=len(t)
  j=bisect(t,i)
  c+=j==m
  i=t[j%m]+1
  if i==n:break
  s=a[i]
k%=c+1
s=[]
for i in a*k:
  if i in s:s=s[:s.index(i)]
  else:s+=i,
print(*s)

Submission status: Runtime Error

Solution code:
import sys
def main():
	input = sys.stdin.readline
	n,k = map(int,input().split())

	aa = [int(item) for item in input().split()]
	p=[-1]*200000
	b=[0]*n
	bl=0
	
	m=[]
	
	while True:
		for i, a in enumerate(aa):
			if p[a]>=0:
				x=p[a]
				for j in range(x,bl):
					p[b[j]]=-1
				bl=x
			else:
				b[bl]=a
				p[a]=bl
				bl+=1
		m.append(b[0:bl])
		if bl==0:
			break;
	
	print(*m[(k-1)%len(m)])
main()

Submission status: Runtime Error

Solution code:
def main(sample_file = ''):

    """ convenient functions
    # for i, a in enumerate(iterable)
    # q, mod = divmod(a, b)
    # divmod(x, y) returns the tuple (x//y, x%y)
    # Higher-order function: reduce(operator.mul, xyz_count, 1)
    # manage median(s) using two heapq https://atcoder.jp/contests/abc127/tasks/abc127_f
    """

    """convenient decorator
    # @functools.lru_cache():
    # to facilitate use of recursive function
        # ex:
        # from functools import lru_cache
        # import sys
        # sys.setrecursionlimit(10**9)
        # @lru_cache(maxsize=None)
        # def fib(n):
        #     if n < 2:
        #         return n
        #     return fib(n-1) + fib(n-2)
        # print(fib(1000))
    """
    
    # import numpy as np
    import sys
    sys.setrecursionlimit(10**7)
    from itertools import accumulate, combinations, permutations, product # https://docs.python.org/ja/3/library/itertools.html
    # accumulate() returns iterator! to get list: list(accumulate())
    from math import factorial, ceil, floor, sqrt
    def factorize(n):
        """return the factors of the Arg and count of each factor
        
        Args:
            n (long): number to be resolved into factors
        
        Returns:
            list of tuples: factorize(220) returns [(2, 2), (5, 1), (11, 1)]
        """
        fct = []  # prime factor
        b, e = 2, 0  # base, exponent
        while b * b <= n:
            while n % b == 0:
                n = n // b
                e = e + 1
            if e > 0:
                fct.append((b, e))
            b, e = b + 1, 0
        if n > 1:
            fct.append((n, 1))
        return fct
    def combinations_count(n, r):
        """Return the number of selecting r pieces of items from n kinds of items.
        
        Args:
            n (long): number
            r (long): number
        
        Raises:
            Exception: not defined when n or r is negative
        
        Returns:
            long: number
        """
        # TODO: How should I do when n - r is negative?
        if n < 0 or r < 0:
            raise Exception('combinations_count(n, r) not defined when n or r is negative')
        if n - r < r: r = n - r
        if r < 0: return 0
        if r == 0: return 1
        if r == 1: return n
        numerator = [n - r + k + 1 for k in range(r)]
        denominator = [k + 1 for k in range(r)]
        for p in range(2,r+1):
            pivot = denominator[p - 1]
            if pivot > 1:
                offset = (n - r) % p
                for k in range(p-1,r,p):
                    numerator[k - offset] /= pivot
                    denominator[k] /= pivot
        result = 1
        for k in range(r):
            if numerator[k] > 1:
                result *= int(numerator[k])
        return result
    def combinations_with_replacement_count(n, r):
        """Return the number of selecting r pieces of items from n kinds of items allowing individual elements to be repeated more than once.
        
        Args:
            n (long): number
            r (long): number
        
        Raises:
            Exception: not defined when n or r is negative
        
        Returns:
            long: number
        """
        if n < 0 or r < 0:
            raise Exception('combinations_with_replacement_count(n, r) not defined when n or r is negative')
        elif n == 0:
            return 1
        else:
            return combinations_count(n + r - 1, r)
    from bisect import bisect_left, bisect_right
    from collections import deque, Counter, defaultdict # https://docs.python.org/ja/3/library/collections.html#collections.deque
    from heapq import heapify, heappop, heappush, heappushpop, heapreplace,nlargest,nsmallest # https://docs.python.org/ja/3/library/heapq.html
    from copy import deepcopy, copy # https://docs.python.org/ja/3/library/copy.html
    import operator
    from operator import itemgetter #sort 
    # ex1: List.sort(key=itemgetter(1))
    # ex2: sorted(tuples, key=itemgetter(1,2))
    from functools import reduce, lru_cache
    def chmin(x, y):
        """change minimum
        if x > y, x = y and return (x, True).
        convenient when solving problems of dp[i]
        
        Args:
            x (long): current minimum value
            y (long): potential minimum value
        
        Returns:
            (x, bool): (x, True) when updated, else (x, False)
        """
        if x > y:
            x = y
            return (x, True)
        else:
            return (x, False)
    def chmax(x, y):
        """change maximum
        if x < y, x = y and return (x, True).
        convenient when solving problems of dp[i]
        
        Args:
            x (long): current maximum value
            y (long): potential maximum value
        
        Returns:
            (x, bool): (x, True) when updated, else (x, False)
        """
        if x < y:
            x = y
            return (x, True)
        else:
            return (x, False)

    from fractions import gcd # Deprecated since version 3.5: Use math.gcd() instead.
    def gcds(numbers):
        return reduce(gcd, numbers)
    def lcm(x, y):
        return (x * y) // gcd(x, y)
    def lcms(numbers):
        return reduce(lcm, numbers, 1)
    def make_divisors(n, reversed=False):
        """create list of divisors
        
        Args:
            number (int): number from which list of divisors is created
            reversed (bool, optional): ascending order if False. Defaults to False.
        
        Returns:
            list: list of divisors
        """
        divisors = set()
        for i in range(1, int(n**0.5)+1):
            if n % i == 0:
                divisors.add(i)
                divisors.add(n//i)
        return sorted(list(divisors),reverse=reversed)
    
    # first create factorial_list
    # fac_list = mod_factorial_list(n)
    INF = 10 ** 18
    MOD = 10 ** 9 + 7
    modpow = lambda a, n, p = MOD: pow(a, n, p) # Recursive function in python is slow!
    def modinv(a, p = MOD):
        # evaluate reciprocal using Fermat's little theorem:
        # a**(p-1) is identical to 1 (mod p) when a and p is coprime
        return modpow(a, p-2, p)
    def modinv_list(n, p = MOD):
        if n <= 1:
            return [0,1][:n+1]
        else:
            inv_t = [0,1]
            for i in range(2, n+1):
                inv_t += [inv_t[p % i] * (p - int(p / i)) % p]
            return inv_t
    def modfactorial_list(n, p = MOD):
        if n == 0:
            return [1]
        else:
            l = [0] * (n+1)
            tmp = 1
            for i in range(1, n+1):
                tmp = tmp * i % p
                l[i] = tmp
            return l
    def modcomb(n, k, fac_list = [], p = MOD):
        # fac_list = modfactorial_list(100)
        # print(modcomb(100, 5, modfactorial_list(100)))
        from math import factorial
        if n < 0 or k < 0 or n < k: return 0
        if n == 0 or k == 0: return 1
        if len(fac_list) <= n:
            a = factorial(n) % p
            b = factorial(k) % p
            c = factorial(n-k) % p
        else:
            a = fac_list[n]
            b = fac_list[k]
            c = fac_list[n-k]
        return (a * modpow(b, p-2, p) * modpow(c, p-2, p)) % p
    def modadd(a, b, p = MOD):
        return (a + b) % MOD
    def modsub(a, b, p = MOD):
        return (a - b) % p
    def modmul(a, b, p = MOD):
        return ((a % p) * (b % p)) % p
    def moddiv(a, b, p = MOD):
        return modmul(a, modpow(b, p-2, p))

    class UnionFindTree:
        """union find tree class
        TODO: fix this description...
        how to use (example):
        >>  uf = UnionFindTree(N) 
        >>  if uf.find_root(a) == uf.find_root(b):
        >>      do something
        >>  else:
        >>      do something
        >>      uf.unite(a, b)
        """
        def __init__(self, N):
            self.root = [-1] * (N+1)
            self.rank   = [0]  * (N+1)
            self.connected_num = [1] * (N+1)
    
        def find_root(self,x):
            root = self.root
            while root[x] != -1:
                x = root[x]
            return x
    
        def unite(self,x,y):
            root = self.root
            rank = self.rank
            connected_num = self.connected_num
            find_root = self.find_root
    
            rx = find_root(x)
            ry = find_root(y)
            if rx != ry:
                if rank[rx] < rank[ry]:
                    root[rx] = ry
                    rx,ry = ry,rx
                else:
                    if rank[rx] == rank[ry]:
                        rank[rx] += 1
                    root[ry] = rx       
                connected_num[rx] += connected_num[ry]

    # Graph: https://en.wikipedia.org/wiki/Directed_graph
    # Bellman-Ford: O(|V||E|). Use this if there exists an edge with negative length in the graph
    # After N steps, the shortest path has converded if there doesn't exist an cycle of edges with negative
    # Watch out: d[N] == d[2*N] doesn't necessarily mean the graph doesn't have negative cycle
    # ref: https://www.youtube.com/watch?v=1Z6ofKN03_Y

    def BellmanFord(N, M, ABC, vertex_start, vertex_end, value_if_inf = -1, find_shortest = False):
        """to calculate furthest or shortest length between vertex_start and vertex_end using BellmanFord algorithm
        
        Args:
            N (int): number of vertices
            M (int): number of edges
            ABC (list): [(ai, bi, ci) for _ in range(N)] where i-th edge is directed from vertex ai to vertex bi and the length is ci 
            vertex_start (int): start vertex. usually use 0.
            vertex_end (int): end vertex. usually use N-1.
            value_if_inf (int or string as you like, optional): value you want when the furthest (or shortest) distance is infinite (or -infinite). Defaults to -1.
            find_shortest (bool, optional): choose False to find furthest path. Defaults to False.
        
        Returns:
            int or string: normally int (but can be str if you set value_if_inf to str)
            
        Example:
                N, M, P = R()
                ABC = [R() for _ in range(M)]
                ABC = [(a-1, b-1, c-P) for a, b, c in ABC]
                print(BellmanFord(N, M, ABC, 0, N-1, value_if_inf = 'inf'))

        """

        def make_reachable_list(N, M, ABC, vertex_start, vertex_end):
            reachable_to_direct = defaultdict(list)
            reachable_from_direct = defaultdict(list)
            reachable_from_start = [False] * N
            reachable_to_end = [False] * N
            reachable_from_start[vertex_start] = True
            reachable_to_end[vertex_end] = True
            reachable_from_both_sides = [False] * N
            dfs_from_start = []
            dfs_to_end = []
            for a, b, c in ABC:
                reachable_to_direct[a].append(b)
                reachable_from_direct[b].append(a)
                if a == vertex_start:
                    dfs_from_start.append(b)
                    reachable_from_start[b] = True
                if b == vertex_end:
                    dfs_to_end.append(a)
                    reachable_to_end[a] = True
            while dfs_from_start:
                v = dfs_from_start.pop()
                for i in reachable_to_direct[v]:
                    if not reachable_from_start[i]:
                        reachable_from_start[i] = True
                        dfs_from_start.append(i)
            while dfs_to_end:
                v = dfs_to_end.pop()
                for i in reachable_from_direct[v]:
                    if not reachable_to_end[i]:
                        reachable_to_end[i] = True
                        dfs_to_end.append(i)
            for i in range(N):
                reachable_from_both_sides[i] = reachable_from_start[i] and reachable_to_end[i]
            return reachable_from_both_sides
        
        reachable_from_both_sides = make_reachable_list(N, M, ABC, vertex_start, vertex_end)
        
        if find_shortest:
            dist = [INF for i in range(N)]
        else:
            dist = [-INF for i in range(N)]
        
        dist[vertex_start] = 0
        for i in range(N):
            updated = False
            for a, b, c in ABC:
                if not reachable_from_both_sides[a]:
                    continue
                elif find_shortest:
                    update_condition = dist[a] + c < dist[b]
                else:
                    update_condition = dist[a] + c > dist[b]
                if dist[a] != INF and update_condition:
                    dist[b] = dist[a] + c
                    updated = True
                    if i == N-1:
                        return value_if_inf
            if not updated:
                break
        return dist[vertex_end]

    """ initialize variables and set inputs
    # initialize variables
        # to initialize list, use [0] * n
        # to initialize two dimentional array, use [[0] * N for _ in range(N)]
    # set inputs
        # put inputs between specific values (almost as quickly)
        # ex) S = [-INF] + [int(r()) for _ in range(A)] + [INF]
        # open(0).read() is sometimes useful:
        # ex) n, m, *x = map(int, open(0).read().split())
        #     min(x[::2]) - max(x[1::2])
        # ex2) *x, = map(int, open(0).read().split())
        #     don't forget to add comma after *x if only one variable is used
    # preprocessing
        # transpose = [x for x in zip(*data)]
        # ex) [[1, 2, 3], [4, 5, 6], [7, 8, 9]] => [(1, 4, 7), (2, 5, 8), (3, 6, 9)]
        # flat = [flatten for inner in data for flatten in inner]
        # ex) [[1, 2, 3], [4, 5, 6], [7, 8, 9]] => [1, 2, 3, 4, 5, 6, 7, 8, 9]
    # calculate and output
        # output pattern
        # ex1) print(*l) => when l = [2, 5, 6], printed 2 5 6
    """

    # functions to read input
    r = lambda: sys.stdin.readline().strip()
    r_int = lambda: int(r())
    R = lambda: list(map(int, r().split()))
    Rmap = lambda: map(int, r().split())
    Rfloat = lambda: list(map(float, r().split()))
    Rtuple = lambda: tuple(map(int, r().split()))

    """ how to treat input
    # single int: int(r())
    # single string: r()
    # single float: float(r())
    # line int: R()
    # line string: r().split()
    # line (str, int, int): [j if i == 0 else int(j) for i, j in enumerate(r().split())]
    # lines int: [R() for _ in range(n)]
    """

    # for test
    if sample_file:
        sys.stdin = open(sample_file)
        
    # ----------------------------------
    # main

    N, K = R()
    A = R()
    dbl = [[0] * N for _ in range(42)]
    pre_num = dict()
    first_num = dict()
    for i, a in enumerate(A):
        if a not in pre_num.keys():
            first_num[a] = i
            pre_num[a] = i
        else:
            dbl[0][pre_num[a]] = (i + 1)%N
            pre_num[a] = i
    for a in pre_num.keys():
        dbl[0][pre_num[a]] = (first_num[a] + 1)%N
    # if not next page
    dbl2 = deepcopy(dbl[0])
    for i in range(N-1, -1, -1):
        if dbl[0][i] >= i + 2:
            dbl[0][i] = dbl[0][dbl[0][i]]

    for i in range(1, 42):
        for j in range(N):
            dbl[i][j] = dbl[i-1][dbl[i-1][j]]
    
    bin_str_K = format(K-1, '#42b')
    idx = 0
    for i, digit in enumerate(bin_str_K):
        if int(digit) == 1:
            idx = dbl[42-i-1][idx]
    
    ans = []
    while idx < N:
        if (first_num[A[idx]] + 1)%N == dbl2[idx]:
            ans.append(A[idx])
            idx += 1
        else:
            idx = dbl2[idx]
            if idx == 0:
                break
    
    print(*ans)
    

    # to finish code, use return instead of exit()
    # end of main
    # ----------------------------------

    """memo: how to use defaultdict of list
    # initialize
    Dic = defaultdict(list)
    # append / extend
    Dic[x].append(y)
    # three methods for loop: keys(), values(), items()
    for k, v in Dic.items():
    """
    
    """memo: how to solve binary problems
    # to make binary digits text
    >>> a = 5
    >>> bin_str_a = format(a, '#06b')
    >>> print(bin_str_a)
    0b0101 # first 2 strings (='0b') indicates it is binary
    """
    
    """memo: how to solve the problem
    creating simple test/answer
    greed
    simple dp
    graph
    """

if __name__ == '__main__':
    main()