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 LuaJIT (2.0.4) (judge verdict: Wrong Answer)

local mmi, mma = math.min, math.max
local n, m = io.read("*n", "*n")
local edge = {}
local v = {}
local len = {}
for i = 1, n do
  edge[i] = {}
  v[i] = false
  len[i] = false
end
v[1] = 0
len[1] = 0
local evmin = 0
local oddmin = false
for i = 1, m do
  local a, b, c = io.read("*n", "*n", "*n")
  edge[a][b] = c
  edge[b][a] = c
  if a == 1 or b == 1 then
    oddmin = c
  end
end
local rem_edge = {}
local tasks = {1}
for i = 1, n do
  local src = tasks[i]
  for dst, sum in pairs(edge[src]) do
    if not v[dst] then
      edge[dst][src] = nil
      v[dst] = sum - v[src]
      len[dst] = 1 - len[src]
      if len[dst] == 1 then
        oddmin = mmi(oddmin, v[dst])
      else
        evmin = mmi(evmin, v[dst])
      end
      table.insert(tasks, dst)
    else
      table.insert(rem_edge, {src, dst, sum})
      edge[dst][src] = nil
    end
  end
end
if #rem_edge == 0 then
  local diff = evmin - 1
  oddmin = oddmin + diff
  print(mma(0, oddmin))
else
  local valid = true
  local offset = false
  for i = 1, #rem_edge do
    local p1, p2, sum = rem_edge[i][1], rem_edge[i][2], rem_edge[i][3]
    if len[p1] + len[p2] == 1 then
      if v[p1] + v[p2] ~= sum then
        valid = false break
      end
    else
      if len[p1] + len[p2] == 0 then
        if not offset then
          offset = sum - v[p1] - v[p2]
          if offset % 2 == 1 then valid = false break end
          offset = math.floor(offset / 2)
        else
          if v[p1] + v[p2] + offset * 2 ~= sum then
            valid = false break
          end
        end
      else
        if not offset then
          offset = v[p1] + v[p2] - sum
          if offset % 2 == 1 then valid = false break end
          offset = math.floor(offset / 2)
        else
          if v[p1] + v[p2] - offset * 2 ~= sum then
            valid = false break
          end
        end
      end
    end
    if offset then
      if evmin + offset < 1 then valid = false end
      if oddmin - offset < 1 then valid = false end
    end
  end
  print(valid and 1 or 0)
end