#include <iostream>
#include <deque>
#include <vector>
#include <set>
using namespace std;
int const NN=2e6+10;
long long dp[NN],pre[NN],val[NN];
long long sol(deque<int>a,deque<int>b,int k)
{
int n=a.size();
pre[0]=0;
for (int i=0;i<n;i++)
pre[i+1]=pre[i]+b[i];
for (int i=0;i<n;i++)
dp[i]=-1e15;
dp[0]=a[0]-pre[k];
multiset<long long>s;
s.insert(dp[0]+pre[k]);
val[0]=dp[0]+pre[k];
long long mx=dp[0];
for (int i=1;i<n;i++)
{
if (i>=k)
s.erase(s.find(val[i-k]));
long long sm=pre[min(n,i+k)]-pre[i];
dp[i]=mx;
if (s.size())
dp[i]=max(dp[i],*rbegin(s)-pre[i]);
dp[i]=dp[i]+a[i]-sm;
mx=max(mx,dp[i]);
val[i]=dp[i]+pre[min(i+k,n)];
s.insert(val[i]);
}
return mx;
}
long long solve1(int N, int K, deque<int>A, deque<int>B)
{
int n=N;
long long dp[N+10][N+10]={};
long long pre[2*N+10]={};
for (int i=0;i<2*N;i++)
pre[i+1]=pre[i]+B[i%N];
for (int i=0;i<N;i++)
for (int j=0;j<N;j++)
dp[i][j]=-1e15;
long long ans=0;
for (int i=0;i<N;i++)
{
dp[i][i]=A[i]-(pre[i+K]-pre[i]);
ans=max(ans,dp[i][i]);
}
for (int i=0;i<n;i++)
{
for (int j=i;j<n;j++)
{
for (int k=j+1;k<n;k++)
{
int l=k,r=k+K;
l=max(l,j+K);
r=min(r,i+n);
long long sm=0;
if (r>=l)
sm=pre[r]-pre[l];
dp[i][k]=max(dp[i][k],dp[i][j]-sm+A[k]);
ans=max(ans,dp[i][k]);
}
}
}
return ans;
}
long long solve(int N, int K, int *A, int *B)
{
long long ans=0;
int n=N;
deque<int>a,b;
for (int i=0;i<n;i++)
a.push_back(A[i]),b.push_back(B[i]);
for (int i=0;i<n;i++)
{
ans=max(ans,sol(a,b,K));
a.push_back(a.front());
a.pop_front();
b.push_back(b.front());
b.pop_front();
}
long long ans1=solve1(N,K,a,b);
if (ans1<ans)
exit(-1);
return ans;
}
