A12306. 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.

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 $a_1, a_2, \dots, a_n$ ( $1 \le a_i \le 10^9$ , $a_i < a_{i+1}$ ) — positions of cities in the ascending order.

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

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

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 $55$ .