#include <bits/stdc++.h>
using namespace std;
//#define int long long
#define pii pair<int, int>
void answer(vector<signed> C, vector<signed> X, vector<signed> Y);
void dfs(vector<int> &X, vector<int> &Y, int v) {
int u = X[-v - 1];
if (u < -1) {
dfs(X, Y, u);
if (X[-u - 1] == -1 && Y[-u - 1] == -1) X[-v - 1] = -1;
}
u = Y[-v - 1];
if (u < -1) {
dfs(X, Y, u);
if (X[-u - 1] == -1 && Y[-u - 1] == -1) Y[-v - 1] = -1;
}
}
void create_circuit6(int k, vector<int> a) {
a.push_back(0);
int n = a.size();
vector<int> C(k + 1, -1), X, Y;
int cnt = n;
int z = 1, ord = 0;
while (z < cnt) {
z *= 2;
ord++;
}
ord = max(1, ord);
vector<int> p(z);
p[0] = 0;
for (int i = 0; i < ord; i++) {
for (int ni = 0; ni < (1 << i); ni++) {
p[(1 << i) + ni] = p[ni] + (1 << (ord - i - 1));
}
}
for (int i = 0; i < z - n; i++) p[i] = -1;
vector<pii> mp(z);
for (int i = 0; i < z; i++) mp[i] = {p[i], i};
sort(mp.begin(), mp.end());
for (int i = z - n; i < z; i++) {
p[mp[i].second] = i - z + n;
}
vector<int> v(z);
for (int i = 0; i < z; i++) {
v[i] = (p[i] == -1 ? -1 : a[p[i]]);
}
vector<int> X_(z), Y_(z);
for (int ni = 1; ni < z / 2; ni++) {
X_[ni] = -ni * 2 - 1;
Y_[ni] = -ni * 2;
}
for (int ni = 0; ni < z / 2; ni++) {
X_[z - ni - 1] = v[ni * 2];
Y_[z - ni - 1] = v[ni * 2 + 1];
}
for (int i = 1; i < z; i++) {
X.push_back(X_[i]);
Y.push_back(Y_[i]);
}
dfs(X, Y, -1);
/*for (int x : X) cout << x << " ";
cout << "\n";
for (int x : Y) cout << x << " ";
cout << "\n\n";*/
vector<int> Xf, Yf, f(z);
int tm = -1;
for (int i = 0; i < z - 1; i++) {
if (X[i] == -1 && Y[i] == -1) continue;
f[i] = tm;
tm--;
}
/*for (int i = 0; i < z; i++) {
cout << f[i] << " ";
}
cout << "\n\n";*/
for (int i = 0; i < z; i++) {
if (X[i] == -1 && Y[i] == -1) continue;
Xf.push_back(X[i] < 0 ? f[-X[i] - 1] : X[i]);
Yf.push_back(Y[i] < 0 ? f[-Y[i] - 1] : Y[i]);
}
/*for (int x : Xf) cout << x << " ";
cout << "\n";
for (int x : Yf) cout << x << " ";
cout << "\n";*/
answer(C, Xf, Yf);
}
void create_circuit3(int k, vector<int> a) {
int n = a.size();
a.push_back(0);
vector<vector<int>> g(k + 1);
g[0].push_back(a[0]);
for (int i = 0; i < n; i++) {
g[a[i]].push_back(a[i + 1]);
}
vector<int> C(k + 1), X, Y;
C[0] = g[0][0];
for (int j = 1; j < k + 1; j++) {
int cnt = g[j].size();
int z = 1, ord = 0;
while (z < cnt) {
z *= 2;
ord++;
}
if (ord == 0) {
if (cnt == 1) C[j] = g[j][0];
continue;
}
vector<int> p(z);
p[0] = 0;
for (int i = 0; i < ord; i++) {
for (int ni = 0; ni < (1 << i); ni++) {
p[(1 << i) + ni] = p[ni] + (1 << (ord - i - 1));
}
}
int wnots = X.size();
C[j] = -wnots - 1;
vector<int> v(z, -wnots - 1);
for (int i = z - (int) g[j].size(); i < z; i++) {
v[i] = g[j][i - z + (int) g[j].size()];
}
vector<int> vp(z);
for (int i = 0; i < z; i++) {
vp[p[i]] = v[i];
}
vector<int> X_(z), Y_(z);
for (int ni = 1; ni < z / 2; ni++) {
X_[ni] = -ni * 2 - wnots;
Y_[ni] = -ni * 2 - 1 - wnots;
}
for (int ni = 0; ni < z / 2; ni++) {
X_[ni + z / 2] = vp[ni * 2];
Y_[ni + z / 2] = vp[ni * 2 + 1];
}
for (int i = 1; i < z; i++) {
X.push_back(X_[i]);
Y.push_back(Y_[i]);
}
}
answer(C, X, Y);
}
void create_circuit(int k, vector<int> a) {
map<int, int> freq;
for (int x : a) {
freq[x]++;
}
for (auto [_, cnt] : freq) {
if (cnt > 4) {
create_circuit6(k, a);
return;
}
}
create_circuit3(k, a);
}
