gold_Milk_Exchange — USACO February_Contest
Task type: batch | Time limit: 2 s | Memory limit: 256 MB | Difficulty: 19
Algorithms: data structures, prefix sum, stack

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

Starter code: