The Hard Work of Paparazzi

题意翻译

曼哈顿有 $r$ 条南北向的街道，按从西到东的顺序用数字表示为 $1 , 2 , ... , r$ ，有 $r$ 条东西向的街道，按从南到北的顺序用数字表示为 $1$ , $2$ , $...$ , $r$ 。每条南北向的街道与每条东西向的街道相交，第 $x$ 条南北向道路与第 $y$ 条东西向道路相交的路口用 $(x,y)$ 表示。从路口 $(x,y)$ 移动到路口 $(x',y')$ 需要 $|x-x'|+|y-y'|$ 分钟。 你知道城市有将有 $n$ 位名人出现并且你想尽可能多地给他们拍照。更确切地说，你知道第 $i$ 位名人将恰好在第 $t_i$ 分钟出现在路口 $(x_i,y_i)$ 并只停留相当短的时间，只有当你第 $t_i$ 分钟时在路口才能拍照。你的工作技法娴熟，所以你能瞬间完成拍照。你已经知道了对任意的$i=1,2,\cdots,n-1$ 都有 $t_i<t_{i+1}$ 成立。 输入的第一行是两个正整数 $r$ , $n$ 表示道路的数量和名人的数量，接下来的n行描述名人的出现，第 $i$ 行有三个正整数 $t_i$ , $x_i$ , $y_i$ 表示第 $i$ 位名人将在第 $t_i$ 分钟出现在路口 $(x_i,y_i)$ 。 输出一个整数，为你能拍到的名人的最大数量。

题目描述

You are a paparazzi working in Manhattan. Manhattan has $ r $ south-to-north streets, denoted by numbers $ 1, 2,\ldots, r $ in order from west to east, and $ r $ west-to-east streets, denoted by numbers $ 1,2,\ldots,r $ in order from south to north. Each of the $ r $ south-to-north streets intersects each of the $ r $ west-to-east streets; the intersection between the $ x $ -th south-to-north street and the $ y $ -th west-to-east street is denoted by $ (x, y) $ . In order to move from the intersection $ (x,y) $ to the intersection $ (x', y') $ you need $ |x-x'|+|y-y'| $ minutes. You know about the presence of $ n $ celebrities in the city and you want to take photos of as many of them as possible. More precisely, for each $ i=1,\dots, n $ , you know that the $ i $ -th celebrity will be at the intersection $ (x_i, y_i) $ in exactly $ t_i $ minutes from now (and he will stay there for a very short time, so you may take a photo of him only if at the $ t_i $ -th minute from now you are at the intersection $ (x_i, y_i) $ ). You are very good at your job, so you are able to take photos instantaneously. You know that $ t_i < t_{i+1} $ for any $ i=1,2,\ldots, n-1 $ . Currently you are at your office, which is located at the intersection $ (1, 1) $ . If you plan your working day optimally, what is the maximum number of celebrities you can take a photo of?

输入输出格式

输入格式

The first line of the input contains two positive integers $ r, n $ ( $ 1\le r\le 500 $ , $ 1\le n\le 100,000 $ ) – the number of south-to-north/west-to-east streets and the number of celebrities. Then $ n $ lines follow, each describing the appearance of a celebrity. The $ i $ -th of these lines contains $ 3 $ positive integers $ t_i, x_i, y_i $ ( $ 1\le t_i\le 1,000,000 $ , $ 1\le x_i, y_i\le r $ ) — denoting that the $ i $ -th celebrity will appear at the intersection $ (x_i, y_i) $ in $ t_i $ minutes from now. It is guaranteed that $ t_i<t_{i+1} $ for any $ i=1,2,\ldots, n-1 $ .

输出格式

Print a single integer, the maximum number of celebrities you can take a photo of.

输入输出样例

输入样例 #1

10 1
11 6 8

输出样例 #1

0

输入样例 #2

6 9
1 2 6
7 5 1
8 5 5
10 3 1
12 4 4
13 6 2
17 6 6
20 1 4
21 5 4

输出样例 #2

4

输入样例 #3

10 4
1 2 1
5 10 9
13 8 8
15 9 9

输出样例 #3

1

输入样例 #4

500 10
69 477 122
73 186 235
341 101 145
372 77 497
390 117 440
494 471 37
522 300 498
682 149 379
821 486 359
855 157 386

输出样例 #4

3

说明

Explanation of the first testcase: There is only one celebrity in the city, and he will be at intersection $ (6,8) $ exactly $ 11 $ minutes after the beginning of the working day. Since you are initially at $ (1,1) $ and you need $ |1-6|+|1-8|=5+7=12 $ minutes to reach $ (6,8) $ you cannot take a photo of the celebrity. Thus you cannot get any photo and the answer is $ 0 $ . Explanation of the second testcase: One way to take $ 4 $ photos (which is the maximum possible) is to take photos of celebrities with indexes $ 3, 5, 7, 9 $ (see the image for a visualization of the strategy): - To move from the office at $ (1,1) $ to the intersection $ (5,5) $ you need $ |1-5|+|1-5|=4+4=8 $ minutes, so you arrive at minute $ 8 $ and you are just in time to take a photo of celebrity $ 3 $ . - Then, just after you have taken a photo of celebrity $ 3 $ , you move toward the intersection $ (4,4) $ . You need $ |5-4|+|5-4|=1+1=2 $ minutes to go there, so you arrive at minute $ 8+2=10 $ and you wait until minute $ 12 $ , when celebrity $ 5 $ appears. - Then, just after you have taken a photo of celebrity $ 5 $ , you go to the intersection $ (6,6) $ . You need $ |4-6|+|4-6|=2+2=4 $ minutes to go there, so you arrive at minute $ 12+4=16 $ and you wait until minute $ 17 $ , when celebrity $ 7 $ appears. - Then, just after you have taken a photo of celebrity $ 7 $ , you go to the intersection $ (5,4) $ . You need $ |6-5|+|6-4|=1+2=3 $ minutes to go there, so you arrive at minute $ 17+3=20 $ and you wait until minute $ 21 $ to take a photo of celebrity $ 9 $ . Explanation of the third testcase: The only way to take $ 1 $ photo (which is the maximum possible) is to take a photo of the celebrity with index $ 1 $ (since $ |2-1|+|1-1|=1 $ , you can be at intersection $ (2,1) $ after exactly one minute, hence you are just in time to take a photo of celebrity $ 1 $ ). Explanation of the fourth testcase: One way to take $ 3 $ photos (which is the maximum possible) is to take photos of celebrities with indexes $ 3, 8, 10 $ : - To move from the office at $ (1,1) $ to the intersection $ (101,145) $ you need $ |1-101|+|1-145|=100+144=244 $ minutes, so you can manage to be there when the celebrity $ 3 $ appears (at minute $ 341 $ ). - Then, just after you have taken a photo of celebrity $ 3 $ , you move toward the intersection $ (149,379) $ . You need $ |101-149|+|145-379|=282 $ minutes to go there, so you arrive at minute $ 341+282=623 $ and you wait until minute $ 682 $ , when celebrity $ 8 $ appears. - Then, just after you have taken a photo of celebrity $ 8 $ , you go to the intersection $ (157,386) $ . You need $ |149-157|+|379-386|=8+7=15 $ minutes to go there, so you arrive at minute $ 682+15=697 $ and you wait until minute $ 855 $ to take a photo of celebrity $ 10 $ .