CF1994E Wooden Game

You are given a forest of $ k $ rooted trees $ ^{\text{∗}} $ . Lumberjack Timofey wants to cut down the entire forest by applying the following operation: - Select a subtree $ ^{\text{†}} $ of any vertex of one of the trees and remove it from the tree. Timofey loves bitwise operations, so he wants the [bitwise OR](https://en.wikipedia.org/wiki/Bitwise_operation#OR) of the sizes of the subtrees he removed to be maximum. Help him and find the maximum result he can obtain. $ ^{\text{∗}} $ A tree is a connected graph without cycles, loops, or multiple edges. In a rooted tree, a selected vertex is called a root. A forest is a collection of one or more trees. $ ^{\text{†}} $ The subtree of a vertex $ v $ is the set of vertices for which $ v $ lies on the shortest path from this vertex to the root, including $ v $ itself.

Each test consists of multiple test cases. The first line contains an integer $ t $ ( $ 1 \leq t \leq 10^4 $ ) — the number of test cases. Then follows the description of the test cases. The first line of each test case contains a single integer $ k $ ( $ 1 \leq k \leq 10^6 $ ) — the number of trees in the forest. This is followed by a description of each of the $ k $ trees: The first line contains a single integer $ n $ ( $ 1 \leq n \leq 10^6 $ ) — the size of the tree. The vertices of the tree are numbered with integers from $ 1 $ to $ n $ . The root of the tree is vertex number $ 1 $ . The second line contains $ n - 1 $ integers $ p_2, p_3, \ldots p_n $ ( $ 1 \leq p_i < i $ ), where $ p_i $ — the parent of vertex $ i $ . It is guaranteed that the sum of $ k $ and $ n $ for all sets of input data does not exceed $ 10^6 $ .

For each test case, output a single integer — the maximum result that can be obtained.

In the second test case, the trees look like this:  The first operation removes the entire second tree.  The second operation removes vertex $ 4 $ from the first tree.  The third operation removes the first tree. The result is $ 6|1|3 = 7 $ ( $ | $ denotes bitwise OR). In the third test case, the entire tree needs to be removed.