human
Can you provide a Python program to find the number of candies he gave out, or tell him that he must have made a mistake given the following:
Solve in Python:
After six days, professor GukiZ decided to give more candies to his students. Like last time, he has $N$ students, numbered $1$ through $N$. Let's denote the number of candies GukiZ gave to the $i$-th student by $p_i$. As GukiZ has a lot of students, he does not remember all the exact numbers of candies he gave to the students. He only remembers the following properties of the sequence $p$:
- The numbers of candies given to each of the first $K$ students ($p_1, p_2, \dots, p_K$) are known exactly.
- All elements of the sequence $p$ are distinct and positive.
- GukiZ didn't give more than $x$ candies to any student (the maximum value in the sequence $p$ is not greater than $x$).
- For each student $i$, there is at least one other student $j$ such that $|p_i - p_j| \le D$.
- The professor gave out the biggest possible total number of candies, i.e. $S = p_1 + p_2 + p_3 + \ldots + p_N$ is maximum possible.
GukiZ would like to know the total number of candies $S$ he had at the beginning. However, times change and after six days, the professor is really tired, so it is possible that there is no sequence $p$ which satisfies the constraints. Can you help GukiZ find the number of candies he gave out, or tell him that he must have made a mistake?

-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first line of each test case contains four space-separated integers $N$, $K$, $x$, $D$.
- The second line contains $K$ distinct space-separated integers $p_1, p_2, \dots, p_K$.

-----Output-----
For each test case, print a single line containing one integer — the number of candies GukiZ had, or $-1$ if there is no valid sequence $p$.

-----Constraints-----
- $1 \le T \le 50$
- $3 \le N \le 10^9$
- $1 \le K \le \mathrm{min}(N, 2 \cdot 10^4)$ 
- $1 \le x \le 2 \cdot 10^9$
- $1 \le D \le 10^9$
- $1 \le p_i \le x$ for each valid $i$
- All values $p_i$ from input are distinct

-----Subtasks-----
Subtask #1 (15 points): $1...
assistant
#Here is Python program to find the number of candies he gave out, or tell him that he must have made a mistake.

try:
# https://www.codechef.com/LTIME63B/problems/GHMC
# Finally.... I properly understood what needs to be done.

    def ctlt(arr, val):
        # find number of values in sorted arr < val
        if arr[0] >= val: return 0
        lo = 0
        hi = len(arr)
        while hi-lo > 1:
            md = (hi+lo)//2
            if arr[md]<val:
                lo = md
            else:
                hi = md

        return hi

    for _ in range(int(input())):
        n,k,x,d = map(int, input().split())
        z = input().strip().split()
        if k > 0:
            ps = list(map(int,z[:k]))
        else:
            ps = [x]

        ps.sort()

        if x<n or x<ps[-1] or n<k:
            print(-1)
            continue

        valchecked = 0
        fillval = 0
        valsdone = False
        isolbelow = True
        lastp = ps[0]

        while not valsdone and n>=k:
            if n == k:
                lo = x+d+1 # put out of range
            else:
            # find best maxfill (before val support)
                lo = 1
                hi = x+1
                while hi-lo>1:
                    md = (hi+lo)//2
                    v = (x-md+1) + ctlt(ps,md)
                    if v<n:
                        hi = md
                    else:
                        lo = md
        
            valsdone = True
            checkto = ctlt(ps,lo)-1
            if checkto >= valchecked:            
            # support all vals
                for p in ps[valchecked+1:checkto+1]:
                    if lastp+d >= p:
                        isolbelow = False
                    elif isolbelow:
                        valsdone = False
                        fillval += lastp+d
                        n -= 1
                        isolbelow = (p > lastp + 2*d )
                    else:
                        isolbelow = True
                    lastp = p
                valchecked = checkto
                if valsdone and isolbelow: 
                # check gap to maxfill
                    if lastp...