Wrong Answer

Consider the following problem: given an array $$$a$$$ containing $$$n$$$ integers (indexed from $$$0$$$ to $$$n-1$$$), find $$$\max\limits_{0 \leq l \leq r \leq n-1} \sum\limits_{l \leq i \leq r} (r-l+1) \cdot a_i$$$. In this problem, $$$1 \leq n \leq 2\,000$$$ and $$$|a_i| \leq 10^6$$$.

In an attempt to solve the problem described, Alice quickly came up with a blazing-fast greedy algorithm and coded it. Her implementation in pseudocode is as follows:

function find_answer(n, a)
    # Assumes n is an integer between 1 and 2000, inclusive
    # Assumes a is a list containing n integers: a[0], a[1], ..., a[n-1]
    res = 0
    cur = 0
    k = -1
    for i = 0 to i = n-1
        cur = cur + a[i]
        if cur < 0
            cur = 0
            k = i
        res = max(res, (i-k)*cur)
    return res

Also, as you can see, Alice's idea is not entirely correct. For example, suppose $$$n = 4$$$ and $$$a = [6, -8, 7, -42]$$$. Then, find_answer(n, a) would return $$$7$$$, but the correct answer is $$$3 \cdot (6-8+7) = 15$$$.

You told Alice that her solution is incorrect, but she did not believe what you said.

Given an integer $$$k$$$, you are to find any sequence $$$a$$$ of $$$n$$$ integers such that the correct answer and the answer produced by Alice's algorithm differ by exactly $$$k$$$. Note that although the choice of $$$n$$$ and the content of the sequence is yours, you must still follow the constraints earlier given: that $$$1 \leq n \leq 2\,000$$$ and that the absolute value of each element does not exceed $$$10^6$$$. If there is no such sequence, determine so.