Problem p03111 - Generation 2

Orig Description

Score : 300 points
Problem StatementYou have N bamboos. The lengths (in centimeters) of these are l_1, l_2, ..., l_N, respectively.
Your objective is to use some of these bamboos (possibly all) to obtain three bamboos of length A, B, C. For that, you can use the following three kinds of magics any number:
Extension Magic: Consumes 1 MP (magic point). Choose one bamboo and increase its length by 1.
Shortening Magic: Consumes 1 MP. Choose one bamboo of length at least 2 and decrease its length by 1.
Composition Magic: Consumes 10 MP. Choose two bamboos and combine them into one bamboo. The length of this new bamboo is equal to the sum of the lengths of the two bamboos combined. (Afterwards, further magics can be used on this bamboo.)
At least how much MP is needed to achieve the objective?
Constraints
3 \leq N \leq 8
1 \leq C < B < A \leq 1000
1 \leq l_i \leq 1000
All values in input are integers.
InputInput is given from Standard Input in the following format:
N A B C
l_1
l_2
:
l_N
OutputPrint the minimum amount of MP needed to achieve the objective.
Sample Input 15 100 90 80
98
40
30
21
80
Sample Output 123
We are obtaining three bamboos of lengths 100, 90, 80 from five bamboos 98, 40, 30, 21, 80. We already have a bamboo of length 80, and we can obtain bamboos of lengths 100, 90 by using the magics as follows at the total cost of 23 MP, which is optimal.
Use Extension Magic twice on the bamboo of length 98 to obtain a bamboo of length 100. (MP consumed: 2)
Use Composition Magic on the bamboos of lengths 40, 30 to obtain a bamboo of length 70. (MP consumed: 10)
Use Shortening Magic once on the bamboo of length 21 to obtain a bamboo of length 20. (MP consumed: 1)
Use Composition Magic on the bamboo of length 70 obtained in step 2 and the bamboo of length 20 obtained in step 3 to obtain a bamboo of length 90. (MP consumed: 10)
Sample Input 28 100 90 80
100
100
90
90
90
80
80
80
Sample Output 20
If we already have all bamboos of the desired lengths, the amount of MP needed is 0. As seen here, we do not necessarily need to use all the bamboos.
Sample Input 38 1000 800 100
300
333
400
444
500
555
600
666
Sample Output 3243

Extracted Specification

Canonicalized Input Description:

An integer N (3 ≤ N ≤ 8), representing some quantity or size.
Three integers A, B, C (1 ≤ C < B < A ≤ 1000).
N integers l_1, l_2, ..., l_N, where each integer is between 1 and 1000.

### Example Input:

```
5
100 90 80
98
40
30
21
80
```

### Function Signature:
Write a function f(N, A, B, C, lengths) that takes in the input.
```python
def f(N: int, A: int, B: int, C: int, lengths: List[int]):
    '''
    N: an integer
    A: an integer
    B: an integer
    C: an integer
    lengths: a list of integers
    '''
```

Test Case Generator

Sure, I'll write a test case generator for the given problem. This generator will create random values for \( N \), \( A \), \( B \), \( C \), and the lengths of \( N \) bamboos. Here is the implementation:

```python
import random

def tcgen():
    # Constraints
    N = random.randint(3, 8)
    
    # Ensure 1 ≤ C < B < A ≤ 1000
    C = random.randint(1, 333)
    B = random.randint(C + 1, 666)
    A = random.randint(B + 1, 1000)

    # Generate lengths of bamboos within the range 1 ≤ l_i ≤ 1000
    bamboo_lengths = [random.randint(1, 1000) for _ in range(N)]

    # Output the test case
    print(N, A, B, C)
    for length in bamboo_lengths:
        print(length)

# Example usage
tcgen()
```

### Explanation
- **N**: Randomly chosen between 3 and 8 (inclusive) because \( 3 \leq N \leq 8 \).
- **C**: Randomly chosen between 1 and 333.
- **B**: Randomly chosen between \( C+1 \) and 666 to ensure \( C < B \).
- **A**: Randomly chosen between \( B+1 \) and 1000 to ensure \( B < A \).
- **bamboo_lengths**: List of \( N \) random integers, each between 1 and 1000 (inclusive).

This script ensures that all the constraints are satisfied while generating the test case. You can run the `tcgen()` function to get random test cases for the problem.

Extract Arguments

def extract_arguments(fh):
    N, A, B, C = map(int, fh.readline().strip().split())
    lengths = [int(fh.readline().strip()) for _ in range(N)]
    return N, A, B, C, lengths