| At last, Mr. X has managed to return to the school at which he formerly |
| taught, with the intention of confronting his suspicious successor, Mr. Y. |
| However, it appears that Mr. Y has fortified his position — he has brainwashed |
| all of his students into believing that he's the world's best teacher through |
| the timeless ploy of replacing classes with extra recesses. There's no way |
| that Mr. Y can be thrown out of the school without first compromising his |
| students' support! |
|
|
| There are **N** students in Mr. Y's class, with IDs from 1 to **N**. In an |
| attempt to maximize security, Mr. Y has arranged them into a hierarchical |
| structure, with each student _i_ either reporting to a commanding student |
| **Ci**, or not reporting to any other student (indicated by **Ci** = 0). |
| There's exactly one student _i_ for whom **Ci** = 0, who is the class leader. |
|
|
| A _chain of command_ is a sequence of one or more students going up the |
| hierarchical structure starting from some student _i_ and ending somewhere |
| between _i_ and the class leader (inclusive): _i_ → **Ci** → **CCi** → etc. |
| It's guaranteed that, for each student _i_, there exists a _chain of command_ |
| beginning at them and ending at the class leader. |
|
|
| Initially, all **N** students are under Mr. Y's control. However, Mr. X is |
| about to perform some bribery of his own. One by one, in order from 1 to |
| **N**, Mr. X will bribe each student with healthy snacks. After the first _b_ |
| bribes, students 1.._b_ will be under Mr. X's control instead of Mr. Y's. |
|
|
| After each of the **N** bribes, Mr. X would like to evaluate the vulnerability |
| of Mr. Y's class to a potential takeover. To do so, he'll determine **N** |
| hypothetical values: for each student _i_, he'll compute the maximum length |
| that a _controlled chain_ beginning with student _i_ could possibly have if 0 |
| or more _promotions_ were to first take place (or 0 if no such chain could |
| exist). |
|
|
| A _controlled chain_ is a _chain of command_ consisting exclusively of |
| students under Mr. X's control. A _promotion_ is a modification to the class |
| structure in which Mr. X selects a certain student _j_ with a commanding |
| student _c_ = **Cj** (such that _c_ ≠ 0 and _c_ ≠ _i_), expels student _c_ |
| (removing them from the class entirely), and brings _j_ up to occupy _c_'s |
| former place (setting **Cj** to **Cc**, and for each other student _k_ such |
| that **Ck** = _c_, now setting **Ck** to j). Note that each of these |
| hypothetical values should be considered independently of the others; Mr. X |
| will never actually perform any _promotions_ and permanently alter the class |
| structure. |
|
|
| In order to reduce the size of the output, these **N2** values should be |
| aggregated into a single integer as follows: Letting **Sb** be the sum of the |
| **N** students' maximum _controlled chain_ lengths after the first _b_ |
| students have been bribed, output (**S1** * **S2** * ... * **SN**) modulo |
| 1,000,000,007. |
|
|
| ### Input |
|
|
| Input begins with an integer **T**, the number of times Mr. X needs to wrest |
| control from Mr. Y. For each time, there is first a line containing the |
| integer **N**. Then, **N** lines follow, the _i_th of which contains the |
| integer **Ci**. |
|
|
| ### Output |
|
|
| For the _i_th time, print a line containing "Case #_i_: " followed by a single |
| integer, the product of all **N** sums of maximum _controlled chain_ lengths, |
| modulo 1,000,000,007. |
|
|
| ### Constraints |
|
|
| 1 ≤ **T** ≤ 80 |
| 1 ≤ **N** ≤ 800,000 |
| 0 ≤ **Ci** ≤ **N** |
|
|
| ### Explanation of Sample |
|
|
| In the first case, after the first student has been bribed, a length-1 |
| _controlled chain_ can begin at student 1, while no _controlled chain_ can |
| begin at student 2. Once the second student has also been bribed, a length-1 |
| _controlled chain_ can still begin at student 1, while a length-2 _controlled |
| chain_ may now begin at student 2 (2 → 1). This results in a total answer of |
| (1 + 0) * (1 + 2) = 3 (modulo 1,000,000,007). |
|
|
| In the second case, after the first two students have been bribed, a length-2 |
| _controlled chain_ can begin at student 1 (1 → 2) if student 1 gets promoted |
| (replacing student 3). The total answer comes out to (1 + 0 + 0) * (2 + 1 + 0) |
| * (3 + 1 + 2) = 18 (modulo 1,000,000,007). |
|
|
| In the third case, after the first two students have been bribed, a length-2 |
| _controlled chain_ may begin at student 1 (1 → 2) if student 2 gets promoted |
| (replacing student 3), and a length-1 _controlled chain_ can similarly begin |
| at student 2 (2 → 1) if student 1 gets promoted (replacing student 3). The |
| total answer comes out to (1 + 0 + 0) * (2 + 2 + 0) * (2 + 2 + 1) = 20 (modulo |
| 1,000,000,007). _ |
|
|
| In the fourth case, the total answer is (1 + 0 + 0 + 0 + 0) * (2 + 2 + 0 + 0 + |
| 0) * (2 + 3 + 2 + 0 + 0) * (2 + 3 + 2 + 1 + 0) * (2 + 3 + 2 + 1 + 3) = 2464 |
| (modulo 1,000,000,007). |
|
|
| In the fifth case, the total answer is 1 * 3 * 7 * 11 * 14 * 16 * 20 * 28 * 33 |
| * 39 * 44 * 47 = 790446393 (modulo 1,000,000,007). |
|
|
|
|
