Problem:
Score : 100 points

Problem Statement

You are given an undirected graph G. G has N vertices and M edges. The vertices are numbered from 1 through N, and the i-th edge (1 ≤ i ≤ M) connects Vertex a_i and b_i. G does not have self-loops and multiple edges.

You can repeatedly perform the operation of adding an edge between two vertices. However, G must not have self-loops or multiple edges as the result. Also, if Vertex 1 and 2 are connected directly or indirectly by edges, your body will be exposed to a voltage of 1000000007 volts. This must also be avoided.

Under these conditions, at most how many edges can you add? Note that Vertex 1 and 2 are never connected directly or indirectly in the beginning.

Constraints

2 ≤ N ≤ 10^5

0 ≤ M ≤ 10^5

1 ≤ a_i < b_i ≤ N

All pairs (a_i, b_i) are distinct.

Vertex 1 and 2 in G are not connected directly or indirectly by edges.

Input

Input is given from Standard Input in the following format:

N M
a_1 b_1
:
a_M b_M

Output

Print the maximum number of edges that can be added.

Sample Input 1

4 1
1 3

Sample Output 1

2

As shown above, two edges can be added. It is not possible to add three or more edges.

Sample Input 2

2 0

Sample Output 2

0

No edge can be added.

Sample Input 3

9 6
1 4
1 8
2 5
3 6
4 8
5 7

Sample Output 3

12

Solution in Ruby (2.3.3) (judge verdict: Wrong Answer)

class FUnionFind
  attr_reader :nodes, :parents, :ranks

  def initialize(n)
    @parents = (0...n).to_a
    @nodes = @parents.map{ |x| [x] }
  end

  def find(i)
    @parents[i]
  end

  def union(i,j)
    i = @parents[i]; j = @parents[j]
    return if i == j
    j,i = i,j if @nodes[i].size < @nodes[j].size
    #p [i,j,@nodes[i],@nodes[j]]
    #p [i,j,@nodes[i].size,@nodes[j].size]
    @nodes[j].each{ |k| @parents[k] = i }
    @nodes[i].concat(@nodes[j])
    @nodes[j].clear
  end
end

N,M = gets.split.map &:to_i
uf = FUnionFind.new(N)
M.times {
  u,v = gets.split.map &:to_i
  uf.union(u-1,v-1)
}

n0,n1,o = 0,0,0
(0...N).each{ |n|
  #p [n, uf.find(n)]
  case uf.find(n)
  when 0; n0 += 1
  when 1; n1 += 1
  else    o  += 1
  end
}
n0,n1 = n1,n0 if n0 > n1
p n0*(n0-1)/2 + (n1+o)*(n1+o-1)/2 - M