Problem:
Score : 900 points

Problem Statement

Given are simple undirected graphs X, Y, Z, with N vertices each and M_1, M_2, M_3 edges, respectively.
The vertices in X, Y, Z are respectively called x_1, x_2, \dots, x_N, y_1, y_2, \dots, y_N, z_1, z_2, \dots, z_N.
The edges in X, Y, Z are respectively (x_{a_i}, x_{b_i}), (y_{c_i}, y_{d_i}), (z_{e_i}, z_{f_i}).

Based on X, Y, Z, we will build another undirected graph W with N^3 vertices.
There are N^3 ways to choose a vertex from each of the graphs X, Y, Z. Each of these choices corresponds to the vertices in W one-to-one. Let (x_i, y_j, z_k) denote the vertex in W corresponding to the choice of x_i, y_j, z_k.

We will span edges in W as follows:

For each edge (x_u, x_v) in X and each w, l, span an edge between (x_u, y_w, z_l) and (x_v, y_w, z_l).

For each edge (y_u, y_v) in Y and each w, l, span an edge between (x_w, y_u, z_l) and (x_w, y_v, z_l).

For each edge (z_u, z_v) in Z and each w, l, span an edge between (x_w, y_l, z_u) and (x_w, y_l, z_v).

Then, let the weight of the vertex (x_i, y_j, z_k) in W be 1,000,000,000,000,000,000^{(i +j + k)} = 10^{18(i + j + k)}. Find the maximum possible total weight of the vertices in an independent set in W, and print that total weight modulo 998,244,353.

Constraints

2 \leq N \leq 100,000

1 \leq M_1, M_2, M_3 \leq 100,000

1 \leq a_i, b_i, c_i, d_i, e_i, f_i \leq N

X, Y, and Z are simple, that is, they have no self-loops and no multiple edges.

Input

Input is given from Standard Input in the following format:

N
M_1
a_1 b_1
a_2 b_2
\vdots
a_{M_1} b_{M_1}
M_2
c_1 d_1
c_2 d_2
\vdots
c_{M_2} d_{M_2}
M_3
e_1 f_1
e_2 f_2
\vdots
e_{M_3} f_{M_3}

Output

Print the maximum possible total weight of an independent set in W, modulo 998,244,353.

Sample Input 1

2
1
1 2
1
1 2
1
1 2

Sample Output 1

46494701

The maximum possible total weight of an independent set is that of the set (x_2, y_1, z_1), (x_1, y_2, z_1), (x_1, y_1, z_2), (x_2, y_2, z_2). The output should be (3 \times 10^{72} + 10^{108}) modulo 998,244,353, which is 46,494,701.

Sample Input 2

3
3
1 3
1 2
3 2
2
2 1
2 3
1
2 1

Sample Output 2

883188316

Sample Input 3

100000
1
1 2
1
99999 100000
1
1 100000

Sample Output 3

318525248

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

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

const int M = 998244353;
const int N = 1e6+7;
const long long B = 1000000000000000000;
typedef long long LL;

LL pb[N];
void pre() {
    pb[0] = 1;
    LL b = B%M;
    for (int i=1; i<N; i++)
        pb[i] = (pb[i-1]*b)%M;
}

vector<int> bfs(vector<vector<int>>adj) {
    int n = adj.size()-1;
    vector<int> col(n+1, N), in(n+1, 0);

    for (int i=n; i>0; i--) {
        bool odd = false, even = false;
        for (int v: adj[i]) {
            if (v < i)  continue;
            if (col[v] == 2)    continue;
            if (col[v] == 1)    odd = true;
            if (col[v] == 0)    even = true;
        }
        if (!even)      col[i] = 0;
        else if (!odd)  col[i] = 1;
        else            col[i] = 2;
    }
    return col;
}

int m[3];
vector<vector<int>> adj[3];
vector<int> dis[3];
LL even[3], odd[3];
int main() {
    pre();

    int n;
    cin>>n;

    for (int g=0; g<3; g++) {
        cin>>m[g];
        adj[g].resize(n+1);
        for (int i=0; i<m[g]; i++) {
            int u, v;
            cin>>u>>v;
            adj[g][u].push_back(v);
            adj[g][v].push_back(u);
        }
        dis[g] = bfs(adj[g]);
        for (int i=1; i<=n; i++)    cout<<dis[g][i]<<" "; cout<<endl;

        for (int i=1; i<=n; i++) {
            if (dis[g][i] == 1)        odd[g] = (odd[g] + pb[i])%M;
            else if (dis[g][i]==0)     even[g] = (even[g] + pb[i])%M;
        }
    }

    LL ans = 0;
    for (int mask = 0; mask < 8; mask ++) {
        if (__builtin_popcount(mask)%2) continue;
        long long cur = 1;
        for (int g=0; g<3; g++)
            if (mask & (1<<g))      cur = (cur * odd[g])%M;
            else                    cur = (cur * even[g])%M;
        ans = (ans + cur)%M;
    }
    cout<<ans<<endl;
}