#include<bits/stdc++.h>
#define ls(x) (x << 1)
#define rs(x) (x << 1 | 1)

using namespace std;
typedef long long LL;
typedef double DB;

const int maxn = 2e6 + 5;

int n, m, len;

LL tmp[maxn];
LL suml[maxn << 2], sumvl[maxn << 2];

LL ansa, ansb, tl, tvl;

int V[maxn], L[maxn], R[maxn];


void build(int x, int l, int r)
{
	if(l == r){
		suml[x] = tmp[l];
		sumvl[x] = 1LL * tmp[l] * l;
		return;
	}
	
	int mid = (l + r) >> 1;
	build(ls(x), l, mid), build(rs(x), mid + 1, r);
	suml[x] = suml[ls(x)] + suml[rs(x)];
	sumvl[x] = sumvl[ls(x)] + sumvl[rs(x)];
}

void update(int x, int l, int r, int pos, int wl, int wr, int fl)
{
	if(l == r){
		suml[x] += wl * fl;
		sumvl[x] += 1LL * l * wl * fl;
		return;
	}
	
	int mid = (l + r) >> 1;
	if(pos <= mid)
		update(ls(x), l, mid, pos, wl, wr, fl);
	else
		update(rs(x), mid + 1, r, pos, wl, wr, fl);
	suml[x] = suml[ls(x)] + suml[rs(x)];
	sumvl[x] = sumvl[ls(x)] + sumvl[rs(x)];
}

void query(int x, int l, int r)
{
	if(l == r){
		ansa = tvl;
		ansb = tl + R[0];
		return;
	}
	
	int mid = (l + r) >> 1;
	LL t = (tvl + sumvl[rs(x)]) - 1LL * (mid + 1) * (tl + suml[rs(x)] + R[0]);
	if(t > 0)
		query(rs(x), mid + 1, r);
	else{
		tvl += sumvl[rs(x)];
		tl += suml[rs(x)];
		query(ls(x), l, mid);
	}
}

void work()
{
	ansa = 0, ansb = 1;
	tl = 0, tvl = 0;
	query(1, 0, len);
	if(!ansb)
		ansa = 0, ansb = 1;
	auto t = __gcd(ansa, ansb);
	ansa /= t, ansb /= t;
	cout << ansa << "/" << ansb << endl;
}

int main()
{
	ios::sync_with_stdio(0);
	cin.tie(0);
	
	cin >> n >> m;
	len = 1e6;
	for(int i = 1; i <= n; ++i)
		cin >> V[i];
	for(int i = 0; i <= n; ++i)
		cin >> L[i];
	for(int i = 0; i <= n; ++i)
		cin >> R[i];
	for(int i = 1; i <= n; ++i)
		tmp[V[i]] += L[i];
	build(1, 0, len);
	
	work();
	while(m--){
		int opt, x, y, z;
		cin >> opt >> x >> y;
		if(opt == 1){
			cin >> z;
			if(x){
				update(1, 0, len, V[x], L[x], R[x], -1);
				L[x] = y, R[x] = z;
				update(1, 0, len, V[x], L[x], R[x], 1);
			}
			else{
				L[x] = y, R[x] = z;
			}
			
		}
		else{
			if(x){
				update(1, 0, len, V[x], L[x], R[x], -1);
				V[x] = y;
				update(1, 0, len, V[x], L[x], R[x], 1);
			}
			else{
				V[x] = y;
			}
		}
		work();
	}
	
	return 0;
}