Problem 2. Milk Exchange

USACO 2024 February Contest, Gold

Farmer John's $N$ $(1 \leq N \leq 5 \cdot 10^5)$ cows are lined up in a circle. The $i$th cow has a bucket with integer capacity $a_i$ $(1 \leq a_i \leq 10^9)$ liters. All buckets are initially full.

Every minute, cow $i$ will pass all the milk in their bucket to cow $i+1$ for $1\le i<N$, with cow $N$ passing its milk to cow $1$. All exchanges happen simultaneously (i.e., if a cow has a full bucket but gives away $x$ liters of milk and also receives $x$ liters, her milk is preserved). If a cow's total milk ever ends up exceeding $a_i$, then the excess milk will be lost.

After each of $1, 2, \dots, N$ minutes, how much total milk is left among all cows?

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains $N$.

The next line contains integers $a_1,a_2,...,a_N$.

OUTPUT FORMAT (print output to the terminal / stdout):

Output $N$ lines, where the $i$-th line is the total milk left among all cows after $i$ minutes.

SAMPLE INPUT:

6
2 2 2 1 2 1

SAMPLE OUTPUT:

8
7
6
6
6
6

Initially, the amount of milk in each bucket is $[2, 2, 2, 1, 2, 1]$.

• After $1$ minute, the amount of milk in each bucket is $[1, 2, 2, 1, 1, 1]$ so the total amount of milk is $8$.
• After $2$ minutes, the amount of milk in each bucket is $[1, 1, 2, 1, 1, 1]$ so the total amount of milk is $7$.
• After $3$ minutes, the amount of milk in each bucket is $[1, 1, 1, 1, 1, 1]$ so the total amount of milk is $6$.
• After $4$ minutes, the amount of milk in each bucket is $[1, 1, 1, 1, 1, 1]$ so the total amount of milk is $6$.
• After $5$ minutes, the amount of milk in each bucket is $[1, 1, 1, 1, 1, 1]$ so the total amount of milk is $6$.
• After $6$ minutes, the amount of milk in each bucket is $[1, 1, 1, 1, 1, 1]$ so the total amount of milk is $6$.

SAMPLE INPUT:

8
3 8 6 4 8 3 8 1

SAMPLE OUTPUT:

25
20
17
14
12
10
8
8

After $1$ minute, the amount of milk in each bucket is $[1, 3, 6, 4, 4, 3, 3, 1]$ so the total amount of milk is $25$.

SAMPLE INPUT:

10
9 9 10 10 6 8 2 1000000000 1000000000 1000000000

SAMPLE OUTPUT:

2000000053
1000000054
56
49
42
35
28
24
20
20

SCORING: Inputs 4-5: $N \le 2000$Inputs 6-8: $a_i \le 2$Inputs 9-13: All $a_i$ are generated uniformly at random in the range $[1,10^9]$. Inputs 14-23: No additional constraints.

Problem credits: Chongtian Ma, Alex Liang, Patrick Deng