P10142 [USACO24JAN] Mooball Teams III P

Farmer John has $N$ cows on his farm ($2 \leq N \leq 2\cdot 10^5$), conveniently numbered $1 \dots N$. Cow $i$ is located at integer coordinates $(x_i, y_i)$ ($1\le x_i,y_i\le N$). Farmer John wants to pick two teams for a game of mooball! One of the teams will be the "red" team; the other team will be the "blue" team. There are only a few requirements for the teams. Neither team can be empty, and each of the $N$ cows must be on at most one team (possibly neither). The only other requirement is due to a unique feature of mooball: an infinitely long net, which must be placed as either a horizontal or vertical line in the plane at a non-integer coordinate, such as $x = 0.5$. FJ must pick teams so that it is possible to separate the teams by a net. The cows are unwilling to move to make this true. Help a farmer out! Compute for Farmer John the number of ways to pick a red team and a blue team satisfying the above requirements, modulo $10^9+7$.

The first line of input contains a single integer $N.$ The next $N$ lines of input each contain two space-separated integers $x_i$ and $y_i$. It is guaranteed that the $x_i$ form a permutation of $1\dots N$, and same for the $y_i$.

A single integer denoting the number of ways to pick a red team and a blue team satisfying the above requirements, modulo $10^9+7$.

##### For Sample 1: We can either choose the red team to be cow 1 and the blue team to be cow 2, or the other way around. In either case, we can separate the two teams by a net (for example, $x=1.5$). ##### For Sample 2: Here are all ten possible ways to place the cows on teams; the $i$th character denotes the team of the $i$th cow, or . if the $i$th cow is not on a team. ``` RRB R.B RB. RBB .RB .BR BRR BR. B.R BBR ``` ##### For Sample 3: Here are all twelve possible ways to place the cows on teams: ``` RRB R.B RBR RB. RBB .RB .BR BRR BR. BRB B.R BBR ``` ##### For Sample 4: Make sure to output the answer modulo $10^9+7$. #### SCORING: - Input 5: $N\le 10$ - Inputs 6-9: $N\le 200$ - Inputs 10-13: $N\le 3000$ - Inputs 14-24: No additional constraints.