A14479 | Interactive Rays

This is an interactive problem.

Your goal is to find a circle on a plane by shooting rays and getting the distance to the circle as a result.

Interactor has three hidden integer parameters that are determined in advance for each test, but which you don't know — $x_c$ , $y_c$ , and $r_c$ . $(x_c, y_c)$ are coordinates of the circle's center and $r_c$ is its radius. The absolute values of $x_c$ , $y_c$ , and $r_c$ do not exceed $10^5$ , and $1 \leq r_c \leq \sqrt{x_c^2 + y_c^2} - 1$ .

You can shoot rays that extend from the origin $(0,0)$ and go via a point $(x_q,y_q)$ with the integer coordinates you specify. For each ray, you get a distance from the ray to the circle or $0$ if the ray intersects the circle.

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The interaction starts with your program printing a query to the standard output and finishes when your program finds and prints the answer to the problem.

Each query has a form of "? $x_q$ $y_q$ ", where $x_q$ and $y_q$ are integers ( $|x_q|, |y_q| \le 10^6$ ; $x_q \ne 0$ or $y_q \ne 0$ ).

The interactor outputs a line with a single floating-point number — the distance between a query ray and a circle that is precise to $10^{-10}$ by an absolute value.

Your program can make the next query, read the output, and so on. You are allowed to do at most $60$ queries. At the end of the interaction, print the answer line "! $x_c$ $y_c$ $r_c$ ", flush the output and exit.

Note, that the output of the interactor is actually rounded to the 10-th digit after a decimal point, so if you are stress-testing your solution locally, make sure that you also perform the corresponding rounding.