English · Original
Chinese · Translation
Formal · Original
While discussing a proper problem A for a Codeforces Round, Kostya created a cyclic array of positive integers $a_1, a_2, \ldots, a_n$. Since the talk was long and not promising, Kostya created a new cyclic array $b_1, b_2, \ldots, b_{n}$ so that $b_i = (a_i \mod a_{i + 1})$, where we take $a_{n+1} = a_1$. Here $mod$ is the [modulo operation](https://en.wikipedia.org/wiki/Modulo_operation). When the talk became interesting, Kostya completely forgot how array $a$ had looked like. Suddenly, he thought that restoring array $a$ from array $b$ would be an interesting problem (unfortunately, not A).
## Input
The first line contains a single integer $n$ ($2 \le n \le 140582$) — the length of the array $a$.
The second line contains $n$ integers $b_1, b_2, \ldots, b_{n}$ ($0 \le b_i \le 187126$).
## Output
If it is possible to restore some array $a$ of length $n$ so that $b_i = a_i \mod a_{(i \mod n) + 1}$ holds for all $i = 1, 2, \ldots, n$, print «_YES_» in the first line and the integers $a_1, a_2, \ldots, a_n$ in the second line. All $a_i$ should satisfy $1 \le a_i \le 10^{18}$. We can show that if an answer exists, then an answer with such constraint exists as well.
It it impossible to restore any valid $a$, print «_NO_» in one line.
You can print each letter in any case (upper or lower).
[samples]
## Note
In the first example:
* $1 \mod 3 = 1$
* $3 \mod 5 = 3$
* $5 \mod 2 = 1$
* $2 \mod 1 = 0$
在讨论 Codeforces 比赛的一个合适的问题 A 时，Kostya 创建了一个正整数的循环数组 $a_1, a_2, dots.h, a_n$。由于讨论冗长且无望，Kostya 创建了一个新的循环数组 $b_1, b_2, dots.h, b_n$，使得 $b_i = (a_i \mod a_{i + 1})$，其中我们取 $a_{n + 1} = a_1$。这里 $\mod$ 是取模运算。当讨论变得有趣时，Kostya 完全忘记了数组 $a$ 原来的样子。突然，他想到从数组 $b$ 恢复数组 $a$ 将会是一个有趣的问题（可惜不是 A）。
第一行包含一个整数 $n$（$2 \leq n \leq 140582$）——数组 $a$ 的长度。
第二行包含 $n$ 个整数 $b_1, b_2, dots.h, b_n$（$0 \leq b_i \leq 187126$）。
如果存在某个长度为 $n$ 的数组 $a$，使得对所有 $i = 1, 2, dots.h, n$ 都满足 $b_i = a_i \mod a_{(i \mod n) + 1}$，则在第一行输出 «_YES_»，并在第二行输出整数 $a_1, a_2, dots.h, a_n$。所有 $a_i$ 应满足 $1 \leq a_i \leq 10^{18}$。我们可以证明，如果存在答案，则一定存在满足此约束的答案。
如果无法恢复任何有效的 $a$，则在一行中输出 «_NO_»。
你可以以任意大小写形式输出每个字母。
## Input
第一行包含一个整数 $n$（$2 \leq n \leq 140582$）——数组 $a$ 的长度。第二行包含 $n$ 个整数 $b_1, b_2, dots.h, b_n$（$0 \leq b_i \leq 187126$）。
## Output
如果存在某个长度为 $n$ 的数组 $a$，使得对所有 $i = 1, 2, dots.h, n$ 都满足 $b_i = a_i \mod a_{(i \mod n) + 1}$，则在第一行输出 «_YES_»，并在第二行输出整数 $a_1, a_2, dots.h, a_n$。所有 $a_i$ 应满足 $1 \leq a_i \leq 10^{18}$。我们可以证明，如果存在答案，则一定存在满足此约束的答案。如果无法恢复任何有效的 $a$，则在一行中输出 «_NO_»。你可以以任意大小写形式输出每个字母。
[samples]
## Note
在第一个例子中：
$1 \mod 3 = 1$
$3 \mod 5 = 3$
$5 \mod 2 = 1$
$2 \mod 1 = 0$
**Definitions**
Let $ n \in \mathbb{Z} $ with $ n \geq 2 $.
Let $ B = (b_1, b_2, \dots, b_n) $ be a given sequence of integers with $ 0 \leq b_i \leq 187126 $.
Let $ A = (a_1, a_2, \dots, a_n) $ be an unknown sequence of positive integers with $ 1 \leq a_i \leq 10^{18} $, satisfying the cyclic constraint:
$$
b_i \equiv a_i \mod a_{i+1}, \quad \text{for all } i \in \{1, \dots, n\},
$$
where indices are taken modulo $ n $, i.e., $ a_{n+1} = a_1 $.
**Constraints**
1. $ 2 \leq n \leq 140582 $
2. $ 0 \leq b_i \leq 187126 $ for all $ i \in \{1, \dots, n\} $
3. $ 1 \leq a_i \leq 10^{18} $ for all $ i \in \{1, \dots, n\} $
4. For each $ i $, $ b_i = a_i \bmod a_{i+1} $, which implies:
$$
a_i = q_i \cdot a_{i+1} + b_i \quad \text{for some integer } q_i \geq 0, \quad \text{and} \quad b_i < a_{i+1}.
$$
**Objective**
Determine whether there exists a sequence $ A = (a_1, \dots, a_n) $ satisfying the above constraints.
- If yes, output "YES" followed by any such valid sequence $ A $.
- If no, output "NO".
API Response (JSON)
{
"problem": {
"name": "E. Restore Array",
"description": {
"content": "While discussing a proper problem A for a Codeforces Round, Kostya created a cyclic array of positive integers $a_1, a_2, \\ldots, a_n$. Since the talk was long and not promising, Kostya created a new ",
"description_type": "Markdown"
},
"platform": "Codeforces",
"limit": {
"time_limit": 2000,
"memory_limit": 262144
},
"difficulty": "None",
"is_remote": true,
"is_sync": true,
"sync_url": null,
"sign": "CF1028E"
},
"statements": [
{
"statement_type": "Markdown",
"content": "While discussing a proper problem A for a Codeforces Round, Kostya created a cyclic array of positive integers $a_1, a_2, \\ldots, a_n$. Since the talk was long and not promising, Kostya created a new ...",
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{
"statement_type": "Markdown",
"content": "在讨论 Codeforces 比赛的一个合适的问题 A 时，Kostya 创建了一个正整数的循环数组 $a_1, a_2, dots.h, a_n$。由于讨论冗长且无望，Kostya 创建了一个新的循环数组 $b_1, b_2, dots.h, b_n$，使得 $b_i = (a_i \\mod a_{i + 1})$，其中我们取 $a_{n + 1} = a_1$。这里 $\\mod$ 是取模运算。...",
"is_translate": true,
"language": "Chinese"
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{
"statement_type": "Markdown",
"content": "**Definitions** \nLet $ n \\in \\mathbb{Z} $ with $ n \\geq 2 $. \nLet $ B = (b_1, b_2, \\dots, b_n) $ be a given sequence of integers with $ 0 \\leq b_i \\leq 187126 $. \nLet $ A = (a_1, a_2, \\dots, a_n) $...",
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"language": "Formal"
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}