You are given an unweighted tree with n vertices. Then n - 1 following operations are applied to the tree. A single operation consists of the following steps:   choose two leaves;  add the length of the simple path between them to the answer;  remove one of the chosen leaves from the tree. 

Initial answer (before applying operations) is 0. Obviously after n - 1 such operations the tree will consist of a single vertex. 

Calculate the maximal possible answer you can achieve, and construct a sequence of operations that allows you to achieve this answer!


-----Input-----

The first line contains one integer number n (2 ≤ n ≤ 2·10^5) — the number of vertices in the tree. 

Next n - 1 lines describe the edges of the tree in form a_{i}, b_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}). It is guaranteed that given graph is a tree.


-----Output-----

In the first line print one integer number — maximal possible answer. 

In the next n - 1 lines print the operations in order of their applying in format a_{i}, b_{i}, c_{i}, where a_{i}, b_{i} — pair of the leaves that are chosen in the current operation (1 ≤ a_{i}, b_{i} ≤ n), c_{i} (1 ≤ c_{i} ≤ n, c_{i} = a_{i} or c_{i} = b_{i}) — choosen leaf that is removed from the tree in the current operation. 

See the examples for better understanding.


-----Examples-----
Input
3
1 2
1 3

Output
3
2 3 3
2 1 1

Input
5
1 2
1 3
2 4
2 5

Output
9
3 5 5
4 3 3
4 1 1
4 2 2

Difficulty: interview

Starter code:

Solutions:
import sys
from collections import deque as dq
n = int(input())

ind = 0
inp = [int(x)-1 for line in sys.stdin.readlines() for x in line.split()]

coupl = [[] for _ in range(n)]
for _ in range(n-1):
    a,b = inp[ind],inp[ind+1]
    ind+=2
    coupl[a].append(b)
    coupl[b].append(a)

Q = dq()
found = [False]*n
master = 0
found[master] = True

dia1 = 0
Q.append(master)
while Q:
    node = Q.popleft()
    dia1 = node
    for nei in coupl[node]:
        if not found[nei]:
            found[nei] = True
            Q.append(nei)


dia2 = 0
Q.append((dia1,0))
dist1 = [0]*n
found = [False]*n
found[dia1] = True
while Q:
    node,d = Q.popleft()
    dia2 = node
    dist1[node]=d
    for nei in coupl[node]:
        if not found[nei]:
            found[nei] = True
            Q.append((nei,d+1))

Q = []
Q.append((dia2,0))
dist2 = [0]*n
found = [False]*n
found[dia2] = True
while Q:
    node,d = Q.pop()
    dist2[node]=d
    for nei in coupl[node]:
        if not found[nei]:
            found[nei] = True
            Q.append((nei,d+1))

neigs = [0]*n

leaves = []
for i in range(n):
    if i != dia1 and i != dia2 and len(coupl[i])==1:
        leaves.append(i)
    neigs[i]=len(coupl[i])
points = 0
lista = []

while leaves:
    node = leaves.pop()
    if dist1[node]<dist2[node]:
        lista.append((dia2,node,node))
        points += dist2[node]
    else:
        lista.append((dia1,node,node))
        points += dist1[node]
    for nei in coupl[node]:
        neigs[nei]-=1
        if neigs[nei]==1:
            leaves.append(nei)
leaves.append(dia2)
while leaves:
    node = leaves.pop()
    lista.append((dia1,node,node))
    points += dist1[node]
    for nei in coupl[node]:
        neigs[nei]-=1
        if neigs[nei]==1:
            leaves.append(nei)
print(points)
for l in lista:
    a,b,c = l
    print(a+1,b+1,c+1)


import sys


def main():
    n = int(input())

    edges = list(map(int, sys.stdin.read().split()))
    tree_edges = dict()
    for i in range(n):
        tree_edges[i + 1] = set()

    for i in range(0, len(edges) - 1, 2):
        tree_edges[edges[i]].add(edges[i + 1])
        tree_edges[edges[i + 1]].add(edges[i])

    init_distants = [-1] * (n + 1)

    queue = [1]
    init_distants[1] = 0

    while queue:
        next_queue = []
        for process in queue:
            for next_vertex in tree_edges[process]:
                if init_distants[next_vertex] == -1:
                    init_distants[next_vertex] = init_distants[process] + 1
                    next_queue.append(next_vertex)
        queue = next_queue

    head = init_distants.index(max(init_distants))
    distants_from_head = [-1] * (n + 1)
    queue = [head]

    distants_from_head[head] = 0

    while queue:
        next_queue = []
        for process in queue:
            for next_vertex in tree_edges[process]:
                if distants_from_head[next_vertex] == -1:
                    distants_from_head[next_vertex] = distants_from_head[process] + 1
                    next_queue.append(next_vertex)
        queue = next_queue

    tail = distants_from_head.index(max(distants_from_head))
    distants_from_tail = [-1] * (n + 1)
    queue = [tail]

    distants_from_tail[tail] = 0

    while queue:
        next_queue = []
        for process in queue:
            for next_vertex in tree_edges[process]:
                if distants_from_tail[next_vertex] == -1:
                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1
                    next_queue.append(next_vertex)
        queue = next_queue

    path_len_sum = 0
    removal_history = list()

    process_queue = []

    for vertex, adj in list(tree_edges.items()):
        if len(adj) == 1:
            process_queue.append(vertex)

    while process_queue:
        next_queue = []

        for leaf in process_queue:

            if leaf == head or leaf == tail:
                continue

            if distants_from_tail[leaf] > distants_from_head[leaf]:
                path_len_sum += distants_from_tail[leaf]
                new_leaves = []

                for w in tree_edges[leaf]:
                    tree_edges[w].remove(leaf)
                    if len(tree_edges[w]) == 1:
                        new_leaves.append(w)
                next_queue.extend(new_leaves)
                removal_history.append("{0} {1} {0}".format(leaf, tail))
            else:
                path_len_sum += distants_from_head[leaf]
                new_leaves = []

                for w in tree_edges[leaf]:
                    tree_edges[w].remove(leaf)
                    if len(tree_edges[w]) == 1:
                        new_leaves.append(w)
                next_queue.extend(new_leaves)
                removal_history.append("{0} {1} {0}".format(leaf, head))
        process_queue = next_queue

    process_queue = [tail]

    while process_queue:
        leaf = process_queue[0]

        if leaf == head:
            continue

        path_len_sum += distants_from_head[leaf]
        new_leaves = []

        for w in tree_edges[leaf]:
            tree_edges[w].remove(leaf)
            if len(tree_edges[w]) == 1:
                new_leaves.append(w)
        process_queue = new_leaves
        removal_history.append("{0} {1} {0}".format(leaf, head))

    print(str(path_len_sum))
    sys.stdout.write("\n".join(removal_history))
    sys.stdout.write("\n")


main()


import sys


def main():
    n = int(input())

    edges = list(map(int, sys.stdin.read().split()))
    tree_edges = dict()
    for i in range(n):
        tree_edges[i + 1] = set()

    for i in range(0, len(edges) - 1, 2):
        tree_edges[edges[i]].add(edges[i + 1])
        tree_edges[edges[i + 1]].add(edges[i])

    init_distants = [-1] * (n + 1)

    queue = [1]
    init_distants[1] = 0

    while queue:
        next_queue = []
        for process in queue:
            for next_vertex in tree_edges[process]:
                if init_distants[next_vertex] == -1:
                    init_distants[next_vertex] = init_distants[process] + 1
                    next_queue.append(next_vertex)
        queue = next_queue

    head = init_distants.index(max(init_distants))
    distants_from_head = [-1] * (n + 1)
    queue = [head]

    distants_from_head[head] = 0

    while queue:
        next_queue = []
        for process in queue:
            for next_vertex in tree_edges[process]:
                if distants_from_head[next_vertex] == -1:
                    distants_from_head[next_vertex] = distants_from_head[process] + 1
                    next_queue.append(next_vertex)
        queue = next_queue

    tail = distants_from_head.index(max(distants_from_head))
    distants_from_tail = [-1] * (n + 1)
    queue = [tail]

    distants_from_tail[tail] = 0

    while queue:
        next_queue = []
        for process in queue:
            for next_vertex in tree_edges[process]:
                if distants_from_tail[next_vertex] == -1:
                    distants_from_tail[next_vertex] = distants_from_tail[process] + 1
                    next_queue.append(next_vertex)
        queue = next_queue

    path_len_sum = 0
    removal_history = list()

    process_queue = []

    for vertex, adj in tree_edges.items():
        if len(adj) == 1:
            process_queue.append(vertex)

    while process_queue:
        next_queue = []

        for leaf in process_queue:

            if leaf == head or leaf == tail:
                continue

            if distants_from_tail[leaf] > distants_from_head[leaf]:
                path_len_sum += distants_from_tail[leaf]
                new_leaves = []

                for w in tree_edges[leaf]:
                    tree_edges[w].remove(leaf)
                    if len(tree_edges[w]) == 1:
                        new_leaves.append(w)
                next_queue.extend(new_leaves)
                removal_history.append("{0} {1} {0}".format(leaf, tail))
            else:
                path_len_sum += distants_from_head[leaf]
                new_leaves = []

                for w in tree_edges[leaf]:
                    tree_edges[w].remove(leaf)
                    if len(tree_edges[w]) == 1:
                        new_leaves.append(w)
                next_queue.extend(new_leaves)
                removal_history.append("{0} {1} {0}".format(leaf, head))
        process_queue = next_queue

    process_queue = [tail]

    while process_queue:
        leaf = process_queue[0]

        if leaf == head:
            continue

        path_len_sum += distants_from_head[leaf]
        new_leaves = []

        for w in tree_edges[leaf]:
            tree_edges[w].remove(leaf)
            if len(tree_edges[w]) == 1:
                new_leaves.append(w)
        process_queue = new_leaves
        removal_history.append("{0} {1} {0}".format(leaf, head))

    print(str(path_len_sum))
    sys.stdout.write("\n".join(removal_history))
    sys.stdout.write("\n")


main()