Ilshat is a software engineer working on efficient data structures. One day he invented a new data structure. This data structure can store a set of non-negative - bit integers, where is a power of two. That is, for some non-negative integer .

The data structure is initially empty. A program using the data structure has to follow the following rules:

- The program can add elements that are -bit integers into the data structure, one at a time, by using the function
`add_element(x)`

. If the program tries to add an element that is already present in the data structure, nothing happens. - After adding the last element the program should call the function
`compile_set()`

exactly once. - Finally, the program may call the function
`check_element(x)`

to check whether the element is present in the data structure. This function may be used multiple times.

When Ilshat first implemented this data structure, he made a bug in the function `compile_set()`

. The bug reorders the binary digits of each element in the set in the same manner. Ilshat wants you to find the exact reordering of digits caused by the bug.

Formally, consider a sequence in which every number from to appears exactly once. We call such a sequence a permutation. Consider an
element of the set, whose digits in binary are (with being the most significant bit). When the function `compile_set()`

is called, this element is replaced by the element .

The same permutation is used to reorder the digits of every element. Any permutation is possible, including the possibility that for each .

For example, suppose that , , and you have inserted into the set integers whose binary representations are `0000`

, `1100`

and `0111`

. Calling the
function `compile_set`

changes these elements to `0000`

, `0101`

and `1110`

, respectively.

Your task is to write a program that finds the permutation by interacting with the data structure. It should (in the following order):

- choose a set of -bit integers,
- insert those integers into the data structure,
- call the function
`compile_set`

to trigger the bug, - check the presence of some elements in the modified set,
- use that information to determine and return the permutation.

Note that your program may call the function `compile_set`

only once.

In addition, there is a limit on the number of times your program calls the library functions. Namely, it may

- call
`add_element`

at most times ( is for "writes"), - call
`check_element`

at most times ( is for "reads").

#### Implementation details

You should implement one function (method):

```
int[] restore_permutation(int n, int w, int r)
```

`n`

: the number of bits in the binary representation of each element of the set (and also the length of ).`w`

: the maximum number of`add_element`

operations your program can perform.`r`

: the maximum number of`check_element`

operations your program can perform.- the function should return the restored permutation .

#### Library functions

In order to interact with the data structure, your program should use the following three functions (methods):

`void add_element(string x)`

This function adds the element described by`x`

to the set.`x`

: a string of`0`

and`1`

characters giving the binary representation of an integer that should be added to the set. The length of`x`

must be .

`void compile_set()`

This function must be called exactly once. Your program cannot call`add_element()`

after this call. Your program cannot call`check_element()`

before this call.`bool check_element(string x)`

This function checks whether the element`x`

is in the modified set.`x`

: a string of`0`

and`1`

characters giving the binary representation of the element that should be checked. The length of`x`

must be .- returns
`true`

if element`x`

is in the modified set, and`false`

otherwise.

Note that if your program violates any of the above restrictions, its grading outcome will be `Wrong Answer`

.

For all the strings, the first character gives the most significant bit of the corresponding integer.

The grader fixes the permutation before the function `restore_permutation`

is called.

#### Example

The grader makes the following function call:

`restore_permutation(4, 16, 16)`

. We have and the program can do at most "writes" and "reads".

The program makes the following function calls:

`add_element("0001")`

`add_element("0011")`

`add_element("0100")`

`compile_set()`

`check_element("0001")`

returns`false`

`check_element("0010")`

returns`true`

`check_element("0100")`

returns`true`

`check_element("1000")`

returns`false`

`check_element("0011")`

returns`false`

`check_element("0101")`

returns`false`

`check_element("1001")`

returns`false`

`check_element("0110")`

returns`false`

`check_element("1010")`

returns`true`

`check_element("1100")`

returns`false`

Only one permutation is consistent with these values returned by `check_element()`

: the permutation . Thus, `restore_permutation`

should return ```
[2,
1, 3, 0]
```

.

#### Subtasks

- (20 points) , , , for at most 2 indices ,
- (18 points) , , ,
- (11 points) , , ,
- (21 points) , , ,
- (30 points) , , .

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