Longest Chain

Time Limit : 8 sec, Memory Limit : 262144 KB

Problem G: Longest Chain

Let us compare two triples a = (xa, ya, za) and b = (xb, yb, zb) by a partial order ∠ defined as follows.

abxa < xb and ya < yb and za < zb

Your mission is to find, in the given set of triples, the longest ascending series a1a2 ∠ ... ∠ ak.

Input

The input is a sequence of datasets, each specifying a set of triples formatted as follows.

m n A B
x1 y1 z1
x2 y2 z2
...
xm ym zm

Here, m, n, A and B in the first line, and all of xi, yi and zi (i = 1, . . . , m) in the following lines are non-negative integers.

Each dataset specifies a set of m + n triples. The triples p1 through pm are explicitly specified in the dataset, the i-th triple pi being (xi, yi, zi). The remaining n triples are specified by parameters A and B given to the following generator routine.

int a = A, b = B, C = ~(1<<31), M = (1<<16)-1;
int r() {
  a = 36969 * (a & M) + (a >> 16);
  b = 18000 * (b & M) + (b >> 16);
  return (C & ((a << 16) + b)) % 1000000;
}

Repeated 3n calls of r() defined as above yield values of xm+1, ym+1, zm+1, xm+2, ym+2, zm+2, ..., xm+n, ym+n, and zm+n, in this order.

You can assume that 1 ≤ m + n ≤ 3 × 105, 1 ≤ A,B ≤ 216, and 0 ≤ xk, yk, zk < 106 for 1 ≤ km + n.

The input ends with a line containing four zeros. The total of m + n for all the datasets does not exceed 2 × 106.

Output

For each dataset, output the length of the longest ascending series of triples in the specified set. If pi1pi2 ∠ ... ∠ pik is the longest, the answer should be k.

Sample Input

6 0 1 1
0 0 0
0 2 2
1 1 1
2 0 2
2 2 0
2 2 2
5 0 1 1
0 0 0
1 1 1
2 2 2
3 3 3
4 4 4
10 0 1 1
3 0 0
2 1 0
2 0 1
1 2 0
1 1 1
1 0 2
0 3 0
0 2 1
0 1 2
0 0 3
0 10 1 1
0 0 0 0

Output for the Sample Input

3
5
1
3

Source: ACM International Collegiate Programming Contest , Asia Regional Aizu, Aizu, Japan, 2013-11-24
http://sparth.u-aizu.ac.jp/icpc2013