Score : $300$ points

Problem Statement

You observed amoebae and kept some records.

Initially, there was one amoeba, numbered $1$.

You made $N$ records. According to the $i$-th record, the amoeba numbered $A_i$ disappeared by dividing itself into two new amoebae, which were then numbered $2i$ and $2i+1$. Here, amoeba $A_i$ is said to be the parent of amoebae $2i$ and $2i+1$.

For each $k=1,\ldots,2N+1$, how many generations away is amoeba $k$ from amoeba $1$?

Constraints

• $1 \leq N \leq 2\times 10^5$
• The records are consistent. That is:- $1\leq A_i \leq 2i-1$.
  • $A_i$ are distinct integers.
• $1\leq A_i \leq 2i-1$.
• $A_i$ are distinct integers.

Input

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

$N$

$A_1$ $A_2$ $\ldots$ $A_N$

Output

Print $2N+1$ lines. The $k$-th line should contain the generation distance between amoeba $1$ and amoeba $k$.

2
1 2

0
1
1
2
2

From amoeba $1$, amoebae $2$ and $3$ were born. From amoeba $2$, amoebae $4$ and $5$ were born.

• Amoeba $1$ is zero generations away from amoeba $1$.
• Amoeba $2$ is one generation away from amoeba $1$.
• Amoeba $3$ is one generation away from amoeba $1$.
• Amoeba $4$ is one generation away from amoeba $2$, and two generations away from amoeba $1$.
• Amoeba $5$ is one generation away from amoeba $2$, and two generations away from amoeba $1$.

4
1 3 5 2

0
1
1
2
2
3
3
2
2