Problem:
Score : 800 points

Problem Statement

There is a grid with R rows and C columns.
We call the cell in the r-th row and c-th column (r，c).

Mr. Takahashi wrote non-negative integers into N of the cells, that is, he wrote a non-negative integer a_i into (r_i，c_i) for each i (1≤i≤N).
After that he fell asleep.

Mr. Aoki found the grid and tries to surprise Mr. Takahashi by writing integers into all remaining cells.
The grid must meet the following conditions to really surprise Mr. Takahashi.

Condition 1: Each cell contains a non-negative integer.

Condition 2: For any 2×2 square formed by cells on the grid, the sum of the top left and bottom right integers must always equal to the sum of the top right and bottom left integers.

Determine whether it is possible to meet those conditions by properly writing integers into all remaining cells.

Constraints

2≤R，C≤10^5

1≤N≤10^5

1≤r_i≤R

1≤c_i≤C

(r_i，c_i) ≠ (r_j，c_j) (i≠j)

a_i is an integer.

0≤a_i≤10^9

Input

The input is given from Standard Input in the following format:

R C
N
r_1 c_1 a_1
r_2 c_2 a_2
:
r_N c_N a_N

Output

Print Yes if it is possible to meet the conditions by properly writing integers into all remaining cells.
Otherwise, print No.

Sample Input 1

2 2
3
1 1 0
1 2 10
2 1 20

Sample Output 1

Yes

You can write integers as follows.

Sample Input 2

2 3
5
1 1 0
1 2 10
1 3 20
2 1 30
2 3 40

Sample Output 2

No

There are two 2×2 squares on the grid, formed by the following cells:

Cells (1，1), (1，2), (2，1) and (2，2)

Cells (1，2), (1，3), (2，2) and (2，3)

You have to write 40 into the empty cell to meet the condition on the left square, but then it does not satisfy the condition on the right square.

Sample Input 3

2 2
3
1 1 20
1 2 10
2 1 0

Sample Output 3

No

You have to write -10 into the empty cell to meet condition 2, but then it does not satisfy condition 1.

Sample Input 4

3 3
4
1 1 0
1 3 10
3 1 10
3 3 20

Sample Output 4

Yes

You can write integers as follows.

Sample Input 5

2 2
4
1 1 0
1 2 10
2 1 30
2 2 20

Sample Output 5

No

All cells already contain a integer and condition 2 is not satisfied.

Solution in C (GCC 5.4.1) (judge verdict: Wrong Answer)

#if 0
cat <<EOF >mistaken-paste
#endif
// thanks for @rsk0315_h4x

#pragma GCC diagnostic ignored "-Wincompatible-pointer-types"
#define _USE_MATH_DEFINES

#include <stdio.h>
#include <stdint.h>
#include <stdlib.h>
#include <string.h>
#include <stdbool.h>
#include <math.h>
#include <time.h>

typedef uint64_t ull;
typedef  int64_t sll;

#define N_MAX 1000000

typedef struct {
	sll a;
	sll b;
} hwll;

typedef struct {
	sll a;
	sll b;
	sll c;
} hwllc;

ull n;

int32_t phwllABcomp (const void *left, const void *right) {
	hwll l = *(hwll*)left;
	hwll r = *(hwll*)right;
	if (l.a < r.a) {
		return -1;
	}
	if (l.a > r.a) {
		return +1;
	}
	if (l.b < r.b) {
		return -1;
	}
	if (l.b > r.b) {
		return +1;
	}
	return 0;
}

int32_t ptriplecomp (const void *left, const void *right) {
	hwllc l = *(hwllc*)left;
	hwllc r = *(hwllc*)right;

	if (l.a < r.a) {
		return -1;
	}
	if (l.a > r.a) {
		return +1;
	}
	if (l.b < r.b) {
		return -1;
	}
	if (l.b > r.b) {
		return +1;
	}
	if (l.c < r.c) {
		return -1;
	}
	if (l.c > r.c) {
		return +1;
	}
	return 0;
}

sll a[N_MAX];
sll b[N_MAX];
sll c[N_MAX];
hwll xy[N_MAX];

hwllc g1[N_MAX], g2[N_MAX];
ull glen1 = 0, glen2 = 0;
ull gin1[N_MAX], gin2[N_MAX];
bool check1[N_MAX], check2[N_MAX];
sll c1[N_MAX], c2[N_MAX];
sll min1 = 0, min2 = 0;

bool dfs1 (sll v) {
	check1[v] = true;

	for (sll i = gin1[v]; i < gin1[v + 1]; i++) {
		sll u = g1[i].b;

		if (check1[u]) {
			if (c1[v] + g1[i].c != c1[u]) return false;
		} else {
			c1[u] = c1[v] + g1[i].c;
			if (c1[u] < min1) min1 = c1[u];
			if (!dfs1(u)) return false;
		}
	}
	return true;
}
bool dfs2 (sll v) {
	check2[v] = true;

	for (sll i = gin2[v]; i < gin2[v + 1]; i++) {
		sll u = g2[i].b;

		if (check2[u]) {
			if (c2[v] + g2[i].c != c2[u]) return false;
		} else {
			c2[u] = c2[v] + g2[i].c;
			if (c2[u] < min2) min2 = c2[u];
			if (!dfs2(u)) return false;
		}
	}
	return true;
}

ull solve () {
	sll i, j;

	for (i = 0; i < n; i++) {
		xy[i].a = a[i];
		xy[i].b = i;
	}
	qsort(xy, n, sizeof(hwll), phwllABcomp);
	for (i = 1; i < n; i++) {
		if (xy[i - 1].a != xy[i].a) continue;

		ull v = xy[i - 1].b, u = xy[i].b;

		hwllc xl = {b[v], b[u], c[u] - c[v]};
		hwllc xr = {b[u], b[v], c[v] - c[u]};
		g1[glen1++] = xl;
		g1[glen1++] = xr;
	}
	qsort(g1, glen1, sizeof(hwllc), ptriplecomp);

	for (i = 0; i < n; i++) {
		xy[i].a = b[i];
		xy[i].b = i;
	}
	qsort(xy, n, sizeof(hwll), phwllABcomp);
	for (i = 1; i < n; i++) {
		if (xy[i - 1].a != xy[i].a) continue;

		ull v = xy[i - 1].b, u = xy[i].b;

		hwllc xl = {a[v], a[u], c[u] - c[v]};
		hwllc xr = {a[u], a[v], c[v] - c[u]};
		g2[glen2++] = xl;
		g2[glen2++] = xr;
	}
	qsort(g2, glen2, sizeof(hwllc), ptriplecomp);

	i = j = 0;
	while (i < n) {
		gin1[i] = j;
		while (j < glen1 && g1[i].a == g1[j].a) j++;
		i++;
	}
	gin1[n] = glen1;
	i = j = 0;
	while (i < n) {
		gin2[i] = j;
		while (j < glen2 && g2[i].a == g2[j].a) j++;
		i++;
	}
	gin2[n] = glen2;

	if (!glen1) goto success;

	if (!dfs1(g1[0].a)) goto fail;
	if (!dfs2(g2[0].a)) goto fail;

	for (i = 0; i < n; i++) {
		if (check1[a[i]]) {
			if (c[i] - c1[b[i]] + min1 < 0) goto fail;
		}
		if (check2[b[i]]) {
			if (c[i] - c2[a[i]] + min2 < 0) goto fail;
		}
	}

	success:
	puts("Yes");
	return 0;

	fail:
	puts("No");
	return 1;
}

int32_t main (void) {
	scanf("%*llu%*llu");
	scanf("%llu", &n);
	for (int i = 0; i < n; i++) {
		scanf("%lld", &a[i]);
		scanf("%lld", &b[i]);
		scanf("%lld", &c[i]);
		a[i]--;
		b[i]--;
	}

	solve();

	return 0;
}