| It's dinner time! A group of **N** Foxen are standing silently in a field, | |
| which can be represented as an infinite number line, patiently waiting for | |
| their meals to make an appearance. The _i_th Fox is standing at position | |
| **Pi**, with no two Foxen standing at the same position. There's also one hole | |
| in the ground at each each integral position on the number line. Each of these | |
| holes is the entrance to a mole's den, and the Foxen know that some of these | |
| delicious critters are bound to show up sooner or later! | |
| A little-known fact about Foxen is that, in addition to having an acute array | |
| of regular senses, they possess a SONAR-like ability to emit imperceptible | |
| sound waves and use them to discern objects at great distances. The _i_th Fox | |
| has tuned their wavelength to a distance of **Ri**, allowing them to only | |
| detect moles which emerge from holes at a distance of exactly **Ri** away from | |
| them (that is, at either position **Pi** \- **Ri** or **Pi** \+ **Ri**). | |
| All of a sudden, some number of moles have just popped up from various holes | |
| all at once! No mole popped up at any Fox's position, no two moles popped up | |
| from the same hole, and every mole was detected by at least one Fox. | |
| Furthermore, each Fox _i_ has determined that there's _exactly_ 1 mole at a | |
| distance of **Ri** away from it (as opposed to there being either 0 or 2 such | |
| moles). | |
| Following this initial event, there's been quite some commotion. Some moles | |
| may have retreated back underground, and some new moles may have emerged, all | |
| in any order. At every point in time, the set of moles on the surface is | |
| subject to all of the same restrictions as before, with one difference: Each | |
| Fox _i_ continues to be sure that _at least_ 1 mole is still present at a | |
| distance of **Ri** away from it, but can no longer determine whether or not | |
| there are perhaps now 2 such moles instead. | |
| After some time of this, the Foxen have decided that they're ready to pounce | |
| and "invite" some of the moles currently on the surface over for dinner. | |
| Unfortunately, they've started to become rather overwhelmed with trying to | |
| keep track of which moles may be on the surface, or even roughly how many of | |
| them there might be. Assuming that the Foxen's initial observations were | |
| correct, and that some unknown amount of time has since gone by with moles | |
| surfacing or departing, please help the Foxen determine the number of | |
| different quantities of moles which could possibly have ended up on the | |
| surface. | |
| If it's impossible for their set of initial observations to have been accurate | |
| in the first place, output -1 instead. | |
| ### Input | |
| Input begins with an integer **T**, the number of different fields. For each | |
| field, there is first a line containing the integer **N**. Then **N** lines | |
| follow, the _i_th of which contains the space-separated integers **Pi** and | |
| **Ri**. | |
| ### Output | |
| For the _i_th field, print a line containing "Case #**i**: " followed by a | |
| single integer, the number of different quantities of moles which could | |
| possibly end up on the surface at any point, or -1 if the Foxen's initial | |
| observations must have been inaccurate. | |
| ### Constraints | |
| 1 ≤ **T** ≤ 30 | |
| 1 ≤ **N** ≤ 5,000 | |
| 0 ≤ **Pi** ≤ 1,000,000,000 | |
| 1 ≤ **Ri** ≤ 1,000,000,000 | |
| ### Explanation of Sample | |
| In the first case, it's possible for there to eventually be 1 mole (at either | |
| position -1 or 1), or 2 moles (at both positions -1 and 1). There can't be 0 | |
| moles due to the restriction that the Fox must detect at least 1 of them, and | |
| there can't be more than 2 moles as they'd have to be at positions which the | |
| Fox is unable to detect. | |
| In the third case, it's impossible for a set of moles to have initially popped | |
| up such that each Fox would have detected _exactly_ one of them. | |