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 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 ith of which contains the integer Ci.
For the ith 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.
1 ≤ T ≤ 80
1 ≤ N ≤ 800,000
0 ≤ Ci ≤ N
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).