CF1101F.Trucks and Cities

There are $n$ cities along the road, which can be represented as a straight line. The $i$-th city is situated at the distance of $a_i$ kilometers from the origin. All cities are situated in the same direction from the origin. There are $m$ trucks travelling from one city to another.

Each truck can be described by $4$ integers: starting city $s_i$, finishing city $f_i$, fuel consumption $c_i$ and number of possible refuelings $r_i$. The $i$-th truck will spend $c_i$ litres of fuel per one kilometer.

When a truck arrives in some city, it can be refueled (but refueling is impossible in the middle of nowhere). The $i$-th truck can be refueled at most $r_i$ times. Each refueling makes truck's gas-tank full. All trucks start with full gas-tank.

All trucks will have gas-tanks of the same size $V$ litres. You should find minimum possible $V$ such that all trucks can reach their destinations without refueling more times than allowed.

道路上有 $n$ 座城市，这些城市可被表示为一条直线。第 $i$ 座城市位于距原点 $a_i$ 千米处。所有城市均位于原点的同一侧。有 $m$ 辆卡车在城市之间行驶。

每辆卡车可用 $4$ 个整数描述：起始城市 $s_i$、终点城市 $f_i$、单位距离油耗 $c_i$（升/千米）以及最多允许的加油次数 $r_i$。第 $i$ 辆卡车每行驶一千米消耗 $c_i$ 升燃油。

当一辆卡车抵达某座城市时，它可以进行加油（但在荒郊野外无法加油）。第 $i$ 辆卡车最多可加油 $r_i$ 次。每次加油都会将卡车的油箱加满。所有卡车均从满油状态出发。

所有卡车的油箱容量相同，均为 $V$ 升。你需要求出满足以下条件的最小可能的 $V$ 值：所有卡车均能抵达各自的目的地，且加油次数不超过允许上限。

First line contains two integers $n$ and $m$ ($2 \le n \le 400$, $1 \le m \le 250000$) — the number of cities and trucks.

The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^9$, $a_i \lt a_{i+1}$) — positions of cities in the ascending order.

Next $m$ lines contains $4$ integers each. The $i$-th line contains integers $s_i$, $f_i$, $c_i$, $r_i$ ($1 \le s_i \lt f_i \le n$, $1 \le c_i \le 10^9$, $0 \le r_i \le n$) — the description of the $i$-th truck.

第一行包含两个整数 $n$ 和 $m$（$2 \le n \le 400$，$1 \le m \le 250000$）——分别为城市数量与卡车数量。

第二行包含 $n$ 个整数 $a_1, a_2, \dots, a_n$（$1 \le a_i \le 10^9$，且 $a_i \lt a_{i+1}$）——表示城市位置，按升序排列。

接下来 $m$ 行，每行包含 $4$ 个整数。第 $i$ 行包含整数 $s_i$、$f_i$、$c_i$、$r_i$（$1 \le s_i \lt f_i \le n$，$1 \le c_i \le 10^9$，$0 \le r_i \le n$）——描述第 $i$ 辆卡车。

Print the only integer — minimum possible size of gas-tanks $V$ such that all trucks can reach their destinations.

输出唯一的整数——使得所有卡车均能抵达其目的地的油箱容量 $V$ 的最小可能值。

Let's look at queries in details:

1. the $1$-st truck must arrive at position $7$ from $2$ without refuelling, so it needs gas-tank of volume at least $50$.
2. the $2$-nd truck must arrive at position $17$ from $2$ and can be refueled at any city (if it is on the path between starting point and ending point), so it needs gas-tank of volume at least $48$.
3. the $3$-rd truck must arrive at position $14$ from $10$, there is no city between, so it needs gas-tank of volume at least $52$.
4. the $4$-th truck must arrive at position $17$ from $10$ and can be refueled only one time: it's optimal to refuel at $5$-th city (position $14$) so it needs gas-tank of volume at least $40$.
5. the $5$-th truck has the same description, so it also needs gas-tank of volume at least $40$.
6. the $6$-th truck must arrive at position $14$ from $2$ and can be refueled two times: first time in city $2$ or $3$ and second time in city $4$ so it needs gas-tank of volume at least $55$.

我们详细分析各个查询：

1. 第 $1$ 辆卡车必须从位置 $2$ 出发、不加油抵达位置 $7$，因此其油箱容积至少为 $50$。
2. 第 $2$ 辆卡车必须从位置 $2$ 出发抵达位置 $17$，且可在路径上任意城市（即起点与终点之间的城市）加油，因此其油箱容积至少为 $48$。
3. 第 $3$ 辆卡车必须从位置 $10$ 出发抵达位置 $14$，而两者之间无任何城市，因此其油箱容积至少为 $52$。
4. 第 $4$ 辆卡车必须从位置 $10$ 出发抵达位置 $17$，且最多只能加油一次：最优策略是在第 $5$ 个城市（位置 $14$）加油，因此其油箱容积至少为 $40$。
5. 第 $5$ 辆卡车的条件与第 $4$ 辆相同，因此其油箱容积也至少为 $40$。
6. 第 $6$ 辆卡车必须从位置 $2$ 出发抵达位置 $14$，且最多可加油两次：第一次在城市 $2$ 或 $3$，第二次在城市 $4$，因此其油箱容积至少为 $55$。