You are given a forest — an undirected graph with $n$ vertices such that each its connected component is a tree.

The diameter (aka "longest shortest path") of a connected undirected graph is the maximum number of edges in the shortest path between any pair of its vertices.

You task is to add some edges (possibly zero) to the graph so that it becomes a tree and the diameter of the tree is minimal possible.

If there are multiple correct answers, print any of them.


-----Input-----

The first line contains two integers $n$ and $m$ ($1 \le n \le 1000$, $0 \le m \le n - 1$) — the number of vertices of the graph and the number of edges, respectively.

Each of the next $m$ lines contains two integers $v$ and $u$ ($1 \le v, u \le n$, $v \ne u$) — the descriptions of the edges.

It is guaranteed that the given graph is a forest.


-----Output-----

In the first line print the diameter of the resulting tree.

Each of the next $(n - 1) - m$ lines should contain two integers $v$ and $u$ ($1 \le v, u \le n$, $v \ne u$) — the descriptions of the added edges.

The resulting graph should be a tree and its diameter should be minimal possible.

For $m = n - 1$ no edges are added, thus the output consists of a single integer — diameter of the given tree.

If there are multiple correct answers, print any of them.


-----Examples-----
Input
4 2
1 2
2 3

Output
2
4 2

Input
2 0

Output
1
1 2

Input
3 2
1 3
2 3

Output
2



-----Note-----

In the first example adding edges (1, 4) or (3, 4) will lead to a total diameter of 3. Adding edge (2, 4), however, will make it 2.

Edge (1, 2) is the only option you have for the second example. The diameter is 1.

You can't add any edges in the third example. The diameter is already 2.

Difficulty: introductory

Starter code:

Solutions:
import math
n,m=map(int,input().split())
neigh=[]
for i in range(n):
    neigh.append([])
for i in range(m):
    a,b=map(int,input().split())
    neigh[a-1].append(b-1)
    neigh[b-1].append(a-1)
seen=set()
index=[0]*n
diams=[]
trees=0
for i in range(n):
    if i not in seen:
        trees+=1
        index[i]=trees
        seen.add(i)
        layer=[i]
        prev=None
        pars=[None]
        while layer!=[]:
            newlayer=[]
            newpars=[]
            for i in range(len(layer)):
                vert=layer[i]
                par=pars[i]
                for child in neigh[vert]:
                    if child!=par:
                        newlayer.append(child)
                        newpars.append(vert)
                        index[child]=trees
                        seen.add(child)
            prev=layer
            layer=newlayer
            pars=newpars
        far=prev[0]
        layer=[[far]]
        pars=[None]
        prev=None
        while layer!=[]:
            newlayer=[]
            newpars=[]
            for i in range(len(layer)):
                vert=layer[i][-1]
                par=pars[i]
                for child in neigh[vert]:
                    if child!=par:
                        newlayer.append(layer[i]+[child])
                        newpars.append(vert)
            prev=layer
            layer=newlayer
            pars=newpars
        diam=prev[0]
        lent=len(diam)
        mid=diam[lent//2]
        diams.append((lent-1,mid))
diams.sort(reverse=True)
poss=[diams[0][0]]
if len(diams)>1:
    poss.append(math.ceil(diams[0][0]/2)+1+math.ceil(diams[1][0]/2))
if len(diams)>2:
    poss.append(math.ceil(diams[1][0]/2)+2+math.ceil(diams[2][0]/2))
print(max(poss))
cent=diams[0][1]
for i in range(len(diams)-1):
    print(cent+1,diams[i+1][1]+1)

import sys


def dfs(v, d, prev, i):
    nonlocal mid
    nonlocal M
    M[v] = False
    way[d] = v
    if way[d + 1] == 0:
        mid[i] = way[d // 2]
    mx = (d, v)
    for x in E[v]:
        if x != prev:
            mx = max(mx, dfs(x, d + 1, v, i))

    return mx


sys.setrecursionlimit(2000)
n, m = list(map(int, input().split()))

E = [[] for i in range(n + 1)]
for i in range(m):
    a, b = list(map(int, input().split()))
    E[a].append(b)
    E[b].append(a)

mid = [0] * (n + 2)
c = 0
k = 0
way = [0] * (n + 2)

M = [True] * (n + 1)
M[0] = False
i = -1
while True in M:
    i += 1
    idx = M.index(True)
    p1 = dfs(idx, 0, 0, i)[1]
    way = [0] * (n + 2)
    s, p2 = dfs(p1, 0, 0, i)
    if s > c:
        c = s
        k = i
r = []
for j in range(0, i + 1):
    if j == k:
        continue
    r.append((mid[k], mid[j]))
    E[mid[k]].append(mid[j])
    E[mid[j]].append(mid[k])
p1 = dfs(1, 0, 0, n + 1)[1]
s, p2 = dfs(p1, 0, 0, n + 1)
print(s)
for item in r:
    print(item[0], item[1])


import sys
sys.setrecursionlimit(1100)

def dfs1(u,pre):            #find the components
    vis[u] = True
    now.append(u)
    for v in to[u]:
        if v!=pre:dfs1(v,u)

def dfs2(u,pre):            #calulate the distance
    mxdist[u]=dist[u]
    for v in to[u]:
        if v!=pre:
            dist[v] = dist[u]+1
            dfs2(v,u)
            mxdist[u]=max(mxdist[u],mxdist[v])

try:
    lab = 1
    n, m = [int(x) for x in input().split()]
    to = [[] for i in range(n+10)]
    dist = [0 for i in range(n+10)]
    mxdist = [0 for  i in range(n+10)]

    lab = 2
    for i in range(m):
        u,v = [int(x) for x in input().split()]
        to[u].append(v)
        to[v].append(u)

    com=[]
    vis=[False for i in range(n+10)]
    for i in range(1,n+1):
        if vis[i]==False:
            now=[]
            dfs1(i,0)
            com.append(now)

    lab = 3
    ct = []
    mx = 0
    for lis in com:
        tmp = []
        d = 0
        for root in lis:
            for u in lis:dist[u]=mxdist[u]=0
            dfs2(root,0)
            tmp.append((mxdist[root],root))
            d = max( d, sum( sorted([ mxdist[u] for u in to[root] ])[-2:] ) )
            #print(*[mxdist[u] for u in lis])
        mx = max(mx,d)
        #print('d =',d)
        for x in tmp:
            if x[0]==(d+1)//2:
                center = [x[1] for x in tmp if x[0]==(d+1)//2][0]
        ct.append( ((d+1)//2,center) )

    #print(*ct)

    lab = 4
    ct.sort(reverse=True)
    ans = []
    for i in range(1,len(ct)):
        mx = max(mx,ct[i][0]+1+ct[0][0])
        if i>1:mx = max(mx,ct[i][0]+2+ct[1][0])
        ans.append((ct[i][1],ct[0][1]))
    print(mx)
    for p in ans:
        print(*p)
except Exception as e:
    print('error after lable',lab,', type =',e)


import sys
sys.setrecursionlimit(10000)
class Tree():
   def __init__(self, nodes):
     self.root = None

class Node():
  def __init__(self, val):
    self.parent = None
    self.val = val
    self.children = []
  def add(self, child):
    self.children.append(child)
  def get_max_dist(self, parent,dist):
     max_dist = dist
     for i in self.children:
        if i != parent:
           d = i.get_max_dist(self, dist+1)
           if max_dist < d:
              max_dist = d
     return max_dist
  def get_dist(self):
     return self.get_max_dist(self, 0)
  def get_count_ch(self, parent):
     count = len(self.children)
     for i in self.children:
        if i != parent:
           count += i.get_count_ch(self)-1
     return count
  def calc_child(self):
     return (self.get_count_ch(None),len(self.children), self.val)
     
class Forest():
  def __init__(self, count):
    self.nodes = []  
    self.count = count
    self.set_nodes = set()
  def build(self):
    roots = []
    max_dist = []
    #diam = []
    list_root = []
    for i in self.nodes:
       tree =(list([x.get_dist() for x in i]))
       ma = max(tree)
       max_dist.append(ma)
       #diam.append(ma)
       m = i[tree.index(min(tree))]
       roots.append(m)#.val)
    if len(roots) > 1:
       ind = max_dist.index(max(max_dist))
       if len(roots) > 1 and ind != 0:
          roots[0], roots[ind] = roots[ind], roots[0]
    s = set()
    for i in self.nodes:
      for j in i:
        s.add(j.val)
    r = list(set(range(1, n+1))-s)
    
    if len(roots) > 0:# and len(diam) > 0:
       for i in range(1, len(roots)):
          self.add(roots[0].val, roots[i].val)
          list_root.append((roots[0].val, roots[i].val))
       #print(roots + r)
   
       for i in r:
         self.add(roots[0].val, i)
         list_root.append((roots[0].val, i))
    else:
       if len(r) == 1:
          print(0)
          return
       elif len(r) > 1:
          for i in range(1, len(r)):
             self.add(r[0], r[i])
             list_root.append((r[0], r[i]))
    distances = []
    for i in self.nodes[0]:
       dist =(i.get_dist())
       distances.append(dist)
    print(max(distances))
    for i in list_root:
       print(*i)
       
  def add(self, v, u):
    self.set_nodes.add(v)
    self.set_nodes.add(u)
    v_node, v_list = self.find(v)
    u_node, u_list = self.find(u)
    #print(v_node, u_node)
    if v_node == None and u_node == None:
      v_node = Node(v)
      u_node = Node(u)
      v_node.add(u_node)
      u_node.add(v_node)
      self.nodes.append([v_node, u_node])
    elif v_node != None and u_node != None and v_list != u_list:
      v_node.add(u_node)
      u_node.add(v_node)
      v_list += u_list
      self.nodes.remove(u_list)
    elif v_node == None and u_node != None:
      v_node = Node(v)
      u_node.add(v_node)
      v_node.add(u_node)
      u_list.append(v_node)
    elif v_node != None and u_node == None:
      u_node = Node(u)
      v_node.add(u_node)
      u_node.add(v_node)
      v_list.append(u_node)
  def find(self, value):
    for i_list in self.nodes:
      for i in i_list:        
        if i.val == value:
          return i, i_list
    return None, None
n,m = list(map(int,input().split()))
f = Forest(n)
for i in range(m):
  v,u = list(map(int,input().split()))
  f.add(v, u)

#print(f.nodes)
f.build()