//Author: Kevin
#include<bits/stdc++.h>
//#pragma GCC optimize("O2")
using namespace std;
#define ll long long
#define ull unsigned ll
#define pb emplace_back
#define mp make_pair
#define ALL(x) (x).begin(),(x).end()
#define rALL(x) (x).rbegin(),(x).rend()
#define srt(x) sort(ALL(x))
#define rev(x) reverse(ALL(x))
#define rsrt(x) sort(rALL(x))
#define sz(x) (int)(x.size())
#define inf 0x3f3f3f3f
#define pii pair<int,int>
#define lb(v,x) (int)(lower_bound(ALL(v),x)-v.begin())
#define ub(v,x) (int)(upper_bound(ALL(v),x)-v.begin())
#define uni(v) v.resize(unique(ALL(v))-v.begin())
#define longer __int128_t
void die(string S){puts(S.c_str());exit(0);}
mt19937_64 rng(20210448);
int n,q;
ll l[200200],r[200200],v[200200];
ull prior[200200];
ll s1[200200],s2[200200];
int ls[200200],rs[200200];
int siz[200200];
void pushup(int x)
{
	siz[x]=siz[ls[x]]+siz[rs[x]]+1;
	s1[x]=s1[ls[x]]+s1[rs[x]]+l[x];
	s2[x]=s2[ls[x]]+s2[rs[x]]+l[x]*v[x];
}
int merge(int u,int v)
{
	if(!u||!v) return u+v;
	if(prior[u]>prior[v])
	{
		rs[u]=merge(rs[u],v);
		pushup(u);
		return u;
	}
	else
	{
		ls[v]=merge(u,ls[v]);
		pushup(v);
		return v;
	}
}
void split(int x,int ind,int &a,int &b)
{
	if(!x)
	{
		a=b=0;
		return ;
	}
	if(mp(v[x],x)<mp(v[ind],ind))
	{
		a=x;
		split(rs[x],ind,rs[a],b);
		pushup(a);
		return ;
	}
	else
	{
		b=x;
		split(ls[x],ind,a,ls[b]);
		pushup(b);
		return ;
	}
}
void splits(int x,int s,int &a,int &b)
{
	if(!x)
	{
		a=b=0;
		return ;
	}
	if(siz[ls[x]]>=s)
	{
		b=x;
		split(ls[x],s,a,ls[b]);
		pushup(b);
		return ;
	}
	else
	{
		a=x;
		split(rs[x],s-siz[ls[x]]-1,rs[a],b);
		pushup(a);
		return ;
	}
}
pair<ll,ll> query(int x,ll A,ll B)
{
	if(!B)
	{
		if(!x) return mp(0,1);
		if(s1[rs[x]]) return query(rs[x],A,B);
		if(l[x]) return mp(v[x],1);
		return query(ls[x],A,B);
	}
	if(!x) return mp(A,max(B,1ll));
	ll A2=A+s2[rs[x]]+l[x]*v[x],B2=B+s1[rs[x]]+l[x];
	if(A2>=B2*v[x]) return query(rs[x],A,B);
	return query(ls[x],A+s2[rs[x]]+l[x]*v[x],B+s1[rs[x]]+l[x]);
}
int main()
{
	ios_base::sync_with_stdio(false);
	cin.tie(0);
	cout.tie(0);
	cin>>n>>q;
	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++)
	{
		prior[i]=rng();
		siz[i]=1;
	}
	int rt=0;
	for(int i=1;i<=n;i++)
	{
		int a,b;
		split(rt,i,a,b);
		pushup(i);
		rt=merge(a,merge(i,b));
	}
	auto pr=query(rt,0,r[0]);
	ll g=__gcd(pr.first,pr.second);
	cout<<pr.first/g<<"/"<<pr.second/g<<'\n';
	while(q--)
	{
		int op;
		cin>>op;
		if(op==1)
		{
			int x;
			ll l,r;
			cin>>x>>l>>r;
			if(!x)
			{
				::l[0]=l;
				::r[0]=r;
			}
			else
			{
				int A,B,C;
				split(rt,x,A,B);
				splits(B,1,B,C);
				// assert(B==x);
				::l[x]=l;
				::r[x]=r;
				pushup(x);
				rt=merge(A,merge(B,C));
			}
		}
		else
		{
			int x;
			ll v;
			cin>>x>>v;
			int A,B,C;
			split(rt,x,A,B);
			splits(B,1,B,C);
			rt=merge(A,C);
			// assert(B==x);
			::v[x]=v;
			pushup(x);
			split(rt,x,A,C);
			rt=merge(A,merge(B,C));
		}
		auto pr=query(rt,0,r[0]);
		ll g=__gcd(pr.first,pr.second);
		cout<<pr.first/g<<"/"<<pr.second/g<<'\n';
	}
	return 0;
}