Problem:
Score : 700 points

Problem Statement

You are given two sequences a and b, both of length 2N. The i-th elements in a and b are a_i and b_i, respectively.
Using these sequences, Snuke is doing the job of calculating the beauty of pairs of balanced sequences of parentheses (defined below) of length 2N.
The beauty of a pair (s,t) is calculated as follows:

Let X=0.

For each i between 1 and 2N (inclusive), increment X by a_i if s_i = t_i, and increment X by b_i otherwise.

The beauty of (s,t) is the final value of X.

You will be given Q queries. Process them in order.
In the i-th query, update the value of a_{p_i} to x_i, and the value of b_{p_i} to y_i. Then, find the maximum possible beauty of a pair of balanced sequences of parentheses.

In this problem, only the sequences below are defined to be balanced sequences of parentheses.

An empty string

The concatenation of (, s, ) in this order, where s is a balanced sequence of parentheses

The concatenation of s, t in this order, where s and t are balanced sequences of parentheses

Constraints

1 \leq N,Q \leq 10^{5}

-10^{9} \leq a_i,b_i,x_i,y_i \leq 10^{9}

1 \leq p_i \leq 2N

All input values are integers.

Partial Scores

In the test set worth 200 points, N \leq 5 and Q \leq 10.

In the test set worth 300 points, Q = 1.

Input

Input is given from Standard Input in the following format:

N Q
a_1 a_2 ... a_{2N}
b_1 b_2 ... b_{2N}
p_1 x_1 y_1
:
p_Q x_Q y_Q

Output

Print Q lines. The i-th line should contain the response to the i-th query.

Sample Input 1

2 2
1 1 7 3
4 2 3 3
2 4 6
3 2 5

Sample Output 1

15
15

The first query updates a and b to a=(1,4,7,3),b=(4,6,3,3). The maximum possible beauty is 15 for (s,t) =(()(),()()).

The second query updates a and b to a=(1,4,2,3),b=(4,6,5,3). The maximum possible beauty is 15 for (s,t) =(()(),(())).

Sample Input 2

7 7
34 -20 -27 42 44 29 9 11 20 44 27 19 -31 -29
46 -50 -11 20 28 46 12 13 33 -22 -48 -27 35 -17
7 27 34
12 -2 22
4 -50 -12
3 -32 15
8 -7 23
3 -30 11
4 -2 23

Sample Output 2

311
312
260
286
296
292
327

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;

#define INF_LL (int64)1e18
#define INF (int32)1e9
#define REP(i, n) for(int64 i = 0;i < (n);i++)
#define FOR(i, a, b) for(int64 i = (a);i < (b);i++)
#define all(x) x.begin(),x.end()
#define fs first
#define sc second

using int32 = int_fast32_t;
using uint32 = uint_fast32_t;
using int64 = int_fast64_t;
using uint64 = uint_fast64_t;
using PII = pair<int32, int32>;
using PLL = pair<int64, int64>;

const double eps = 1e-10;

template<typename A, typename B>inline void chmin(A &a, B b){if(a > b) a = b;}
template<typename A, typename B>inline void chmax(A &a, B b){if(a < b) a = b;}

template<typename T>
vector<T> make_v(size_t a){return vector<T>(a);}

template<typename T,typename... Ts>
auto make_v(size_t a,Ts... ts){
  return vector<decltype(make_v<T>(ts...))>(a,make_v<T>(ts...));
}

template<typename T,typename U,typename... V>
typename enable_if<is_same<T, U>::value!=0>::type
fill_v(U &u,const V... v){u=U(v...);}

template<typename T,typename U,typename... V>
typename enable_if<is_same<T, U>::value==0>::type
fill_v(U &u,const V... v){
  for(auto &e:u) fill_v<T>(e,v...);
}

template<class ValueMonoid, template<class...> class Container=::std::vector>
class SegTree{
public:
    using value_structure = ValueMonoid;
    using value_type = typename value_structure::value_type;
    using const_reference = const value_type &;
    using container_type = Container<value_type>;
    using size_type = typename container_type::size_type;

private:
    ::std::vector<value_type> tree;
    size_type size_;

    static size_type getsize(const size_type x){
        size_type ret = 1;
        while(ret < x)
            ret <<= 1;
        return ret;
    }

    inline value_type calc(const value_type a, const value_type b){
        return value_structure::operation(a, b);
    }

    inline void calc_node(const size_type index){
        if(tree.size() <= (index << 1 | 1)) return;
        tree[index] = value_structure::operation(tree[index<<1], tree[index<<1 | 1]);
    }
public:
    SegTree() : size_(0), tree(){}
    SegTree(const size_type size)
            : size_(size), tree(size << 1, value_structure::identity()){}
    template<class InputIterator>
    SegTree(InputIterator first, InputIterator last)
            : size_(::std::distance(first, last)){
        tree = container_type(size_, value_structure::identity());
        tree.insert(tree.end(), first, last);
        for(size_type i = size_;i > 0;i--){
            calc_node(i);
        }
    }

    size_type size() const { return size_; }
    const_reference operator[](const size_type k) const {
        assert(k < size_);
        return tree[k+size_];
    }

    value_type query(size_type l, size_type r){
        assert(l <= r);
        assert(0 <= l && l < size_);
        assert(0 <= r && r <= size_);
        value_type retl = value_structure::identity(), retr = value_structure::identity();
        for(l += size_, r += size_; l < r ; l >>= 1, r >>= 1){
            if(l&1) retl = calc(retl, tree[l++]);
            if(r&1) retr = calc(tree[--r], retr);
        }
        return calc(retl, retr);
    }

    template<class F>
    void update(size_type index, const F& f){
        assert(0 <= index && index < size());
        index += size_;
        tree[index] = f(::std::move(tree[index]));
        while(index >>= 1)
            calc_node(index);
    }

    /*
    template<class F>
    size_type search(const F& f) const { // [0, result) is True and [0, result-1) is not.
        if(f(value_structure::identity()))
            return 0;
        if(!f(tree[1]))
            return size_+1;
        value_type acc = value_structure::identity();
        size_type i = 1;
        while(i <
    }
    */
};

class v_monoid {
public:
    using value_type = PLL;
    static value_type identity() { return PLL(-INF_LL, -INF_LL); }
    static value_type initializer() { return {-INF_LL, -INF_LL}; }
    static value_type operation(const value_type& a, const value_type& b) {
        if (a == identity()) return b;
        if (b == identity()) return a;
        return {max(a.fs+b.fs, a.sc+b.sc), max(a.fs+b.sc, a.sc+b.fs)};
    }
};

template <typename T>
struct SegmentTree{
    using F = function<T(T,T)>;
    int n;
    F f;
    T ti;
    vector<T> dat;
    SegmentTree(){};
    SegmentTree(F f,T ti):f(f),ti(ti){}
    void init(int n_){
        n=1;
        while(n<n_) n<<=1;
        dat.assign(n<<1,ti);
    }
    void build(const vector<T> &v){
        int n_=v.size();
        init(n_);
        for(int i=0;i<n_;i++) dat[n+i]=v[i];
        for(int i=n-1;i;i--)
            dat[i]=f(dat[(i<<1)|0],dat[(i<<1)|1]);
    }
    void set_val(int k,T x){
        dat[k+=n]=x;
        while(k>>=1)
            dat[k]=f(dat[(k<<1)|0],dat[(k<<1)|1]);
    }
    T query(int a,int b){
        T vl=ti,vr=ti;
        for(int l=a+n,r=b+n;l<r;l>>=1,r>>=1) {
            if(l&1) vl=f(vl,dat[l++]);
            if(r&1) vr=f(dat[--r],vr);
        }
        return f(vl,vr);
    }
    template<typename C>
    int find(int st,C &check,T &acc,int k,int l,int r){
        if(l+1==r){
            acc=f(acc,dat[k]);
            return check(acc)?k-n:-1;
        }
        int m=(l+r)>>1;
        if(m<=st) return find(st,check,acc,(k<<1)|1,m,r);
        if(st<=l&&!check(f(acc,dat[k]))){
            acc=f(acc,dat[k]);
            return -1;
        }
        int vl=find(st,check,acc,(k<<1)|0,l,m);
        if(~vl) return vl;
        return find(st,check,acc,(k<<1)|1,m,r);
    }
    template<typename C>
    int find(int st,C &check){
        T acc=ti;
        return find(st,check,acc,1,0,n);
    }
};


int main(void) {
    int64 N, Q;
    cin >> N >> Q;
    N *= 2;
    vector<int64> a(N), b(N);
    REP(i, N) cin >> a[i];
    REP(i, N) cin >> b[i];
    int64 res = a[0] + a.back();
    SegTree<v_monoid> sg(N);
    FOR(i, 1, N-1) {
        sg.update(i, [&](const PLL x) { return PLL(a[i], b[i]); });
    }
    REP(i, Q) {
        int64 p, x, y;
        cin >> p >> x >> y; p--;
        if (p == 0 || p == N-1) {
            res -= a[p]; res += x;
            a[p] = x;
        } else {
            sg.update(p, [&](const PLL alskdj) { return PLL(x, y); });
        }
        cout << res + (N > 1 ? sg.query(1, N).fs : 0)<< endl;
    }
}