| As the owner of a vast collection of fine ladders, you would like to display | |
| them to potential onlookers. You've decided to stand your **N** ladders up | |
| vertically, with their bases at distinct points on a horizontal number line. | |
| The **i**th ladder's base is at position **Xi**, and it has height **Hi**. | |
| The local snake population has taken an interest in your ladders. As everyone | |
| knows, snakes love nothing more than to suspend themselves horizontally above | |
| the ground! In particular, a snake of length **L** is able to suspend itself | |
| between the tops of two ladders **a** and **b** if and only if they meet the | |
| following conditions: | |
| * \- the ladders are exactly **L** units apart (|**Xa** \- **Xb**| = **L**) | |
| * \- the ladders are of equal height (**Ha** = **Hb**) | |
| * \- there are no taller ladders in between them (there exists no ladder **c** such that min{**Xa**, **Xb**} < **Xc** < max{**Xa**, **Xb**} and **Hc** > **Ha**) | |
| A number of snakes are planning to take up residence amongst your ladders. In | |
| particular, for every position in which a snake could suspend itself (in other | |
| words, for every distinct, unordered pair of ladders **a** and **b**), one | |
| snake will move in of the appropriate length. | |
| You'll have no choice but to take care of these snakes, of course. To feed a | |
| snake of length **L**, it'll cost you **L**2 dollars daily. To prepare your | |
| budget, you'd like to calculate how many dollars you'll be spending each day | |
| on your new pets! | |
| ### Input | |
| Input begins with an integer **T**, the number of sets of ladders you own. For | |
| each set, there is first a line containing the integer **N**. Then, **N** | |
| lines follow, the **i**th of which contains two space-separated integers, | |
| **Xi** and **Hi**. | |
| ### Output | |
| For the **i**th set of ladders, print a line containing "Case #**i**: " | |
| followed by the daily feeding cost of all the snakes that move in, modulo 109 | |
| \+ 7. | |
| ### Constraints | |
| 1 ≤ **T** ≤ 50 | |
| 1 ≤ **N** ≤ 200,000 | |
| 0 ≤ **Xi**, **Hi** ≤ 1,000,000,000 | |
| ### Explanation of Sample | |
| In the first case, one snake will move in between your two ladders. It will | |
| have length 20, so it will cost 202 = 400 dollars a day to feed. In the third | |
| case, one snake will move in between the ladders of height 3, and another will | |
| move in between the ladders at X = 2 and X = 3. The first snake has length 3, | |
| and the second has length 1. The total cost is therefore 32 \+ 12 = 10. | |