You are given a rooted undirected tree consisting of $$$n$$$ vertices. Vertex $$$1$$$ is the root.

Let's denote a depth array of vertex $$$x$$$ as an infinite sequence $$$[d_{x, 0}, d_{x, 1}, d_{x, 2}, \dots]$$$, where $$$d_{x, i}$$$ is the number of vertices $$$y$$$ such that both conditions hold:

• $$$x$$$ is an ancestor of $$$y$$$;
• the simple path from $$$x$$$ to $$$y$$$ traverses exactly $$$i$$$ edges.

The dominant index of a depth array of vertex $$$x$$$ (or, shortly, the dominant index of vertex $$$x$$$) is an index $$$j$$$ such that:

• for every $$$k < j$$$, $$$d_{x, k} < d_{x, j}$$$;
• for every $$$k > j$$$, $$$d_{x, k} \le d_{x, j}$$$.

For every vertex in the tree calculate its dominant index.