A Bipartite Graph is an undirected graph whose vertices can be partitioned into $2$ sets such that, for each edge $(u, v)$, $u$ and $v$ belong to different sets.
Socket has challenged Bash to a Bipartite Battle. In the Bipartite Battle, Bash and Socket play a game with Bipartite Graphs.
The Bipartite Battle happens as follows:
Socket draws $N$ bipartite graphs. The $i$-th graph has $2$ sets of vertices, one with $a_ i$ vertices, and the other with $b_ i$ vertices.
Then the $2$ players, Bash and Socket alternatively take turns. In each turn, a player must choose exactly one non-empty graph, then delete exactly one edge or exactly one vertex of the chosen graph. If the player deletes one vertex, all edges adjacent to it are also deleted.
The player who cannot move loses. Note that this can only happen when all edges and all vertices of all graphs have been deleted.
Bash plays first.
Of course, Socket does not want to play fair. Socket plans to draw bipartite graphs such that he always wins.
How many ways can Socket draw $N$ bipartite graphs, so that he always wins, assuming both players play optimally?
For the $i$-th bipartite graph, let’s number the vertices in first set from $1$ to $a_ i$, and the vertices in second set from $1$ to $b_ i$.
An edge connecting vertex $u$ in first set and vertex $v$ in second set is denoted as $(u, v)$.
Two drawings of bipartite graphs are considered different iff there exists an index $j$ and a pair of integers $(u, v)$, such that:
$(u, v)$ is an edge of $j$-th graph in one drawing.
$(u, v)$ is NOT an edge of $j$-th graph in the other drawing.
The first line of input contains the integer $N$ $(1 \le N \le 10^5)$.
$N$ lines follow, each line contains exactly $2$ space-separated integers $a_ i$ and $b_ i$ $(1 \le a_ i, b_ i \le 10^9)$.
Print exactly one integer — the number of ways Socket can draw $N$ bipartite graphs, modulo $10^9 + 7$.
|Sample Input 1||Sample Output 1|
1 1 1
|Sample Input 2||Sample Output 2|
1 1 2