A sequence of positive integers $T=(T_1,T_2,\dots,T_M)$ of length $M$ is a palindrome if and only if $T_i=T_{M-i+1}$ for each $i=1,2,\dots,M$.

You are given a sequence of non-negative integers $A = (A_1,A_2,\dots,A_N)$ of length $N$. Determine if there is a sequence of positive integers $S=(S_1,S_2,\dots,S_N)$ of length $N$ that satisfies the following condition, and if it exists, find the lexicographically smallest such sequence.

-   For each $i=1,2,\dots,N$, both of the following hold:
    -   The sequence $(S_{i-A_i},S_{i-A_i+1},\dots,S_{i+A_i})$ is a palindrome.
    -   If $2 \leq i-A_i$ and $i+A_i \leq N-1$, the sequence $(S_{i-A_i-1},S_{i-A_i},\dots,S_{i+A_i+1})$ is not a palindrome.

The input is given from Standard Input in the following format:

```
$N$
$A_1$ $A_2$ $\dots$ $A_N$
```

If there is no sequence $S$ that satisfies the condition, print `No`.

If there is a sequence $S$ that satisfies the condition, let $S'$ be the lexicographically smallest such sequence and print it in the following format:

```
Yes
$S'_1$ $S'_2$ $\dots$ $S'_N$
```

-   $1 \leq N \leq 2 \times 10^5$
-   $0 \leq A_i \leq \min\{i-1,N-i\}$
-   All input values are integers.

Yes
1 1 2 1 1 1 2

S=(1,1,2,1,1,1,2) satisfies the condition:

• $i=1: (A_1)=(1)$ is a palindrome.
• $i=2: (A_2)=(1)$ is a palindrome, but $(A_1,A_2,A_3)=(1,1,2)$ is not.
• $i=3: (A_1,A_2,…,A_5)=(1,1,2,1,1)$ is a palindrome.
• $i=4: (A_4)=(1)$ is a palindrome, but $(A_3,A_4,A_5)=(2,1,1)$ is not.
• $i=5: (A_3,A_4,…,A_7)=(2,1,1,1,2)$ is a palindrome.
• $i=6: (A_6)=(1)$ is a palindrome, but $(A_5,A_6,A_7)=(1,1,2)$ is not.
• $i=7: (A_7)=(2)$ is a palindrome.

There are other sequences like S=(2,2,1,2,2,2,1) that satisfy the condition, but print the lexicographically smallest one, which is (1,1,2,1,1,1,2).