Problem Description

This is an interactive problem.

You are given a positive integer $N$ and a sequence $S=(S_1,S_2,\ldots,S_N)$ of length $N$ whose elements are all either $0$ or $1$.

There is a directed bipartite graph $G$ with $N$ vertices numbered $1,2,\ldots,N$. Each vertex of $G$ is colored: vertex $v$ is white if $S_v=0$ and black if $S_v=1$. Every directed edge $(u,v)$ in $G$ satisfies both of the following conditions:

• $u<v$.
• Vertices $u$ and $v$ have different colors.

The judge has $G$ hidden, and you are not explicitly given any information about its edges. Instead, you may send the following two types of queries to the judge, at most $15000$ queries in total:

• in u V: Ask the judge about a black vertex $u$ and a set $V$ of zero or more white vertices. The judge answers whether there exists a vertex $v$ in $V$ such that the edge $(v,u)$ exists.
• out u V: Ask the judge about a black vertex $u$ and a set $V$ of zero or more white vertices. The judge answers whether there exists a vertex $v$ in $V$ such that the edge $(u,v)$ exists.

Output any one of the longest paths in $G$.

Note that the hidden graph $G$ is fixed before the interaction and remains unchanged.

Interaction

First, an integer $N$ ($2\le N\le 500$) is given from standard input. The next line contains $N$ integers $S_1,S_2,\ldots,S_N$, each of which is $0$ or $1$, separated by spaces.

N
S_1 S_2 ... S_N

After reading the input, you may make at most $15000$ in queries and out queries in total, as described below.

in query

First, choose a black vertex $u$ and a set $V$ of zero or more white vertices. Then, construct a string $T=T_1T_2\ldots T_N$ of length $N$ consisting only of 0s and 1s such that $T_v$ is 1 if and only if $v\in V$. Print the query to standard output in the following format:

in u T

The judge determines whether there exists a vertex $v$ in $V$ such that the edge $(v,u)$ exists. Let $r$ be $1$ if such a vertex exists and $0$ otherwise. The value $r$ is given from standard input in the following format:

r

out query

First, choose a black vertex $u$ and a set $V$ of zero or more white vertices. Then, construct a string $T=T_1T_2\ldots T_N$ of length $N$ consisting only of 0s and 1s such that $T_v$ is 1 if and only if $v\in V$. Print the query to standard output in the following format:

out u T

The judge determines whether there exists a vertex $v$ in $V$ such that the edge $(u,v)$ exists. Let $r$ be $1$ if such a vertex exists and $0$ otherwise. The value $r$ is given from standard input in the following format:

r

Once you have determined a longest path, let $v_1,v_2,\ldots,v_k$ be the vertex numbers in the order in which they appear on the path, and print your answer to standard output in the following format:

! k v_1 v_2 ... v_k

Terminate your program immediately after outputting the answer.

Notes on Interactive Judging

If your output does not conform to the specified format, or if your program terminates without outputting an answer, the verdict is undefined. Terminate your program immediately after outputting the answer. Otherwise, the verdict is undefined.

Whenever you output something, end it with a newline and flush standard output.

If you make more than $15000$ queries, the verdict is undefined. Note that outputting the answer does not count as a query.

5
1 0 0 1 0

1

1

0

1

out 1 01101

in 4 01100

in 4 01000

out 4 01001

! 4 1 3 4 5

Hint

In this sample interaction, the hidden graph has the edge set $E(G)=\{(1,2),(1,3),(1,5),(3,4),(4,5)\}$. This information is not given to the program.