| 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. |
|
|
|
|
