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)
es = {}
M.times {
  u,v,s = gets.split.map(&:to_i)
  es[u] ||= {}; es[v] ||= {}
  es[-u] ||= {}; es[-v] ||= {}
  es[u][-v] = es[-u][v] = es[v][-u] = es[-v][u] = s
}

distance = { 1 => 0 }
constraint = { 1 => 0 }

ns = { 1 => 1 }
d = 0
while ns.size > 0
  d += 1
  nns = {}
  ns.keys.each{ |a|
    c = constraint[a]
    es[a].each{ |b,s|
      next if distance[b]
      distance[b] = d
      if constraint[b]
        if constraint[b] != s - c
          p 0; exit
        end
      else
        constraint[b] = s - c
        nns[b] = 1
      end
    }
  }
  ns = nns
end

vmax = (1..N).max{ |a| constraint[-a] || 0 }; vmin = 0
(1..N).each{ |a|
case
when constraint[a] && constraint[-a]
  #puts "x = (#{constraint[-a]} - #{constraint[a]})/2"
  d = constraint[-a] - constraint[a]
  if d % 2 == 1
    p 0; exit
  end
  vmax = d/2+1
  vmin = d/2-1
when constraint[a]
  #puts "x > #{-constraint[a]}"
  vmin = [vmin, -constraint[a]].max
when distance[-a]
  #puts "x < #{constraint[-a]}"
  vmax = [vmax, constraint[-a]].min
end
}
#p vmin, vmax
p [vmax - vmin - 1, 0].max