You are given a binary string (s) of length (n) and a tree (T) with (n) vertices. Let (k) be the number of 1 s in (s). We will construct a complete undirected weighted graph with (k) vertices as follows: For each (1\le i\le n) with (s_i = \mathtt{1}), create a vertex labeled (i). For any two vertices labeled (u) and (v) that are created in the above step, define the edge weight between them (w(u, v)) as the distance(^{\text{∗}}) between vertex (u) and vertex (v) in the tree (T). A simple path(^{\text{†}}) that visits vertices labeled (v_1, v_2, \ldots, v_m) in this order is nice if for all (1\le i\le m - 2), the condition (2\cdot w(v_i, v_{i + 1})\le w(v_{i + 1}, v_{i + 2})) holds. In other words, the weight of each edge in the path must be at least twice the weight of the previous edge. Note that (s_{v_i} = \mathtt{1}) has to be satisfied for all (1\le i\le m), as otherwise, there would be no vertex with the corresponding label. For each vertex labeled (i) ((1\le i\le n) and (s_i = \mathtt{1})) in the complete undirected weighted graph, determine the maximum number of vertices in any nice simple path starting from the vertex labeled (i). (^{\text{∗}})The distance between two vertices (a) and (b) in a tree is equal to the number of edges on the unique simple path between vertex (a) and vertex (b). (^{\text{†}})A path is a sequence of vertices (v_1, v_2, \ldots, v_m) such that there is an edge between (v_i) and (v_{i + 1}) for all (1\le i\le m - 1). A simple path is a path with no repeated vertices, i.e., (v_i\neq v_j) for all (1\le i < j\le m). Each test contains multiple test cases. The first line contains the number of test cases (t) ((1 \le t \le 10^4)). The description of the test cases follows. The first line of each test case contains a single integer (n) ((1\le n\le 7\cdot10^4)) —