| The market is hot for geological talent! Hiring agents are competing to display their job flyers on street intersections across \(T\) towns. | |
| A town's streets can be represented as an infinite 2D grid with rows numbered by increasing integers from North to South and columns numbered by increasing integers from West to East. | |
| There are \(N\) hiring agents, numbered from \(1\) to \(N\). Agent \(i\) starts at the intersection of row \(R_i\) and column \(C_i\), holding \(P_i\) flyers, and facing the cardinal direction \(D_i\) (`"N"` for North, `"S"` for South, `"E"` for East, or `"W"` for West). Each *turn* taken by an agent consists of the following: | |
| 1. Post one of their own flyers at their current intersection, shredding any competing flyer that may have previously been there. | |
| 2. If the agent has more flyers, move to the adjacent intersection in direction \(D_i\). | |
| The agents will take turns posting one flyer at a time. Agent \(1\) will take a turn, followed by agent \(2\), and so on until agent \(N\) takes a turn, at which point agent \(1\) will go again (if necessary). Note that there may be multiple agents occupying the same intersection at any point in time. | |
| Let \(F_i\) be the number of flyers placed by agent \(i\) which remain will unshredded at the end. Your job is to compute the sum of \(i * F_i\) over all agents \(i\). As this sum may be very large, you should only compute it modulo \(1{,}000{,}000{,}007\). | |
| # Constraints | |
| \(1 \le T \le 75\) | |
| \(1 \le N \le 800{,}000\) | |
| \(0 \le R_i, C_i \le 1{,}000{,}000{,}000\) | |
| \(1 \le P_i \le 1{,}000{,}000{,}000\) | |
| \(D_i \in \{\)`"N"`, `"S"`, `"E"`, `"W"`\(\}\) | |
| The sum of \(N\) across all towns is at most \(2{,}000{,}000\). | |
| # Input | |
| Input begins with an integer \(T\), the number of towns. For each town, there is first a line containing a single integer \(N\). Then, \(N\) lines follow, the \(i\)th of which consists of \(3\) space-separated integers \(R_i\), \(C_i\), and \(P_i\), followed by a space-separated character, \(D_i\). | |
| # Output | |
| For the \(i\)th town, print a line containing *"Case #i: "* followed by a single integer, the sum of \(i * F_i\) over all agents \(i\), modulo \(1{,}000{,}000{,}007\). | |
| # Sample Explanation | |
| In the first town, the relevant portion of the grid will end up with the following flyers (with `"."` representing an intersection with no flyer), yielding an answer of \((1 * 3 + 2 * 2 + 3 * 1 + 4 * 1) \;\mathrm{mod}\; 1{,}000{,}000{,}007 = 14\): | |
| {{PHOTO_ID:1597395337265553|WIDTH:180}} | |
| In the second town, the relevant portion of the grid will end up with the following flyers, yielding an answer of \((1 * 2 + 2 * 2 + 3 * 2 + 4 * 2) \;\mathrm{mod}\; 1{,}000{,}000{,}007 = 20\): | |
| {{PHOTO_ID:4543454065770744|WIDTH:234}} | |
| In the third town, each of agent \(2\)'s turns will involve shredding the last flyer placed by agent \(1\) and replacing it with her own. This will result in \(1{,}000{,}000{,}000\) intersections containing agent \(2\)'s flyers, for an answer of \((2 * 1{,}000{,}000{,}000) \;\mathrm{mod}\; 1{,}000{,}000{,}007 = 999999993\). | |
| In the fourth town, the relevant portion of the grid will end up with the following flyers: | |
| {{PHOTO_ID:451563133198975|WIDTH:360}} | |