Jack and Jill play a game called Hotter, Colder. Jill has a number between 1 and N, and Jack makes repeated attempts to guess it.

Each of Jack's guesses is a number between 1 and N. In response to each guess, Jill answers hotter, colder or same. For Jack's first guess, Jill answers same. For the remaining guesses Jill answers:

hotter if this guess is closer to Jill's number than his previous guess
colder if this guess is farther from Jill's number than his previous guess
same if this guess is neither closer to nor further from Jill's number than his previous guess.

You are to implement a procedure HC(N) that plays Jack's role. This implementation may repeatedly call Guess(G), with G a number between 1 and N. Guess(G) will return 1 to indicate hotter, -1 to indicate colder or 0 to indicate same. HC(N) must return Jill's number.

Example

As example, assume N=5, and Jill has chosen the number 2. When procedure HC makes the following sequence of calls to Guess, the results in the second column will be returned.

Call	Returned value	Explanation
Guess(5)	0	Same (first call)
Guess(3)	1	Hotter
Guess(4)	-1	Colder
Guess(1)	1	Hotter
Guess(3)	0	Same

At this point Jack knows the answer, and HC should return 2. It has taken Jack 5 guesses to determine Jill's number. You can do better.

Subtask 1 [25 points]

HC(N) must call Guess(G) at most 500 times. There will be at most 125 250 calls to HC(N), with N between 1 and 500.

Subtask 2 [25 points]

HC(N) must call Guess(G) at most 18 times. There will be at most 125 250 calls to HC(N) with N between 1 and 500.

Subtask 3 [25 points]

HC(N) must call Guess(G) at most 16 times. There will be at most 125 250 calls to HC(N) with N between 1 and 500.

Subtask 4 [up to 25 points]

Let W be the largest integer, such that 2^W≤3N. For this subtask your solution will score:

0 points, if HC(N) calls Guess(G) 2W times or more,
25α points, where α is the largest real number, such that 0<α<1 and HC(N) calls Guess(G) at most 2W-αW times,
25 points, if HC(N) calls Guess(G) at most W times.

There will be at most 1 000 000 calls to HC(N) with N between 1 and 500 000 000.

Be sure to initialize any variables used by HC every time it is called.

Implementation Details

To be implemented by contestant: hottercolder.c or hottercolder.cpp or hottercolder.pas
Contestant interface: hottercolder.h or hottercolder.pas
Grader interface: grader.h or graderlib.pas
Sample grader: grader.c or grader.cpp or grader.pas and graderlib.pas
Sample grader input: grader.in.1 grader.in.2.
Note: The input file contains several lines, each containing N and Jill's number.
Expected output for sample grader input: the grader will create files grader.out.1 grader.out.2 etc.
- If the implementation correctly implements Subtask 1, one line of output will contain OK 1
- If the implementation correctly implements Subtask 2, one line of output will contain OK 2
- If the implementation correctly implements Subtask 3, one line of output will contain OK 3
- If the implementation correctly implements Subtask 4, one line of output will contain OK 4 alpha α