Problem:
Score : 600 points

Problem Statement

Kenkoooo found a simple connected graph.
The vertices are numbered 1 through n.
The i-th edge connects Vertex u_i and v_i, and has a fixed integer s_i.

Kenkoooo is trying to write a positive integer in each vertex so that the following condition is satisfied:

For every edge i, the sum of the positive integers written in Vertex u_i and v_i is equal to s_i.

Find the number of such ways to write positive integers in the vertices.

Constraints

2 \leq n \leq 10^5

1 \leq m \leq 10^5

1 \leq u_i < v_i \leq n

2 \leq s_i \leq 10^9

If i\neq j, then u_i \neq u_j or v_i \neq v_j.

The graph is connected.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

n m
u_1 v_1 s_1
:
u_m v_m s_m

Output

Print the number of ways to write positive integers in the vertices so that the condition is satisfied.

Sample Input 1

3 3
1 2 3
2 3 5
1 3 4

Sample Output 1

1

The condition will be satisfied if we write 1,2 and 3 in vertices 1,2 and 3, respectively.
There is no other way to satisfy the condition, so the answer is 1.

Sample Input 2

4 3
1 2 6
2 3 7
3 4 5

Sample Output 2

3

Let a,b,c and d be the numbers to write in vertices 1,2,3 and 4, respectively.
There are three quadruples (a,b,c,d) that satisfy the condition:

(a,b,c,d)=(1,5,2,3)

(a,b,c,d)=(2,4,3,2)

(a,b,c,d)=(3,3,4,1)

Sample Input 3

8 7
1 2 1000000000
2 3 2
3 4 1000000000
4 5 2
5 6 1000000000
6 7 2
7 8 1000000000

Sample Output 3

0

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

n, m = gets.split.map(&:to_i)
edges = Array.new(n + 1){|i|{}}
m.times {|i|
  u, v, s = gets.split.map(&:to_i)
  edges[u][v] = s
  edges[v][u] = s
}

nodes = Array.new(n + 1) {|i|
  {
    :index => i,
    :min => 0,
    :max => 1000000000,
    :diff => 1000000000,
    :dirty => false
  }
}

nodes[1][:dirty] = true

queue = []
queue << 1
while queue.length > 0
  u = queue.shift
  node = nodes[u]
  next if !node[:dirty]

  node[:dirty] = false
  o_min = node[:min]
  o_max = node[:max]
  edges[u].each{|v, s|
    min = 1
    max = s - 1
    tn = nodes[v]
    node[:min] = [min, node[:min], s - tn[:max]].max
    node[:max] = [max, node[:max], s - tn[:min]].min
    node[:diff] = node[:max] - node[:min] + 1

    if node[:max] + nodes[v][:max] < s
      puts "0"
      exit
    end
  }
  if node[:min] > node[:max]
    puts "0"
    exit
  end
  next if o_min == node[:min] && o_max == node[:max]

  edges[u].each{|v, s|
    nodes[v][:dirty] = true
    queue << v
  }
end

puts nodes[1][:diff]