You, a young astronomer, are interested in the spatial arrangement of a set of stars with distinctive features. Knowing this might contribute to the understanding of early-universe cosmology. You can perform measurements using a specialized instrument for this purpose.
The instrument uses its own three-dimensional Cartesian coordinate system, in which the origin $$$(0,0,0)$$$ is set on Earth, and the positions of the stars are modeled as lattice points (points whose coordinates are all integers). You already know the distances to all the stars of interest from Earth, but their directions are unknown.
In a single measurement, you specify two distinct stars, and the instrument reports the distance between them. Note that the instrument does not report the absolute or relative directions of the stars.
Determine the distances between all pairs of the stars within the limited number of measurements.
The first line of input contains an integer $$$n,$$$ the number of stars of interest ($$$2 \le n \le 100$$$). The stars are numbered from $$$1$$$ to $$$n.$$$ The second line contains $$$n$$$ integers. The $$$i$$$-th of them is the squared distance from the origin to star $$$i.$$$ It is guaranteed that all stars have integer coordinates, each between $$$-4000$$$ and $$$4000$$$, inclusive. No two stars are at the same position. No star is at the origin.
After reading in these two lines, you may start measurements. To measure the distance between stars $$$i$$$ and $$$j$$$, write a line of the form "measure $$$i$$$ $$$j$$$", where $$$i$$$ and $$$j$$$ are distinct integers between $$$1$$$ and $$$n$$$, inclusive. In response, an input line containing an integer denoting the squared distance between stars $$$i$$$ and $$$j$$$ becomes available. You can perform up to 300 measurements.
When you have determined the distances between all pairs of the stars, write a line containing only answer, followed by $$$n-1$$$ lines of the following format.
| $$$d_{1,2}$$$ $$$d_{1,3}$$$ $$$\cdots$$$ $$$d_{1,n}$$$ |
| $$$d_{2,3}$$$ $$$\cdots$$$ $$$d_{2,n}$$$ |
| $$$\vdots$$$ |
| $$$d_{n-1, n}$$$ |
If your output does not conform to the specifications above, or if the number of measurements exceeds 300, your submission will be judged as a wrong answer.
The coordinates of the stars are fixed before the interaction starts; they do not change during the interaction.
You are provided with a command-line tool for local testing. For more details, refer to the attachments in the contest system.
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In Sample Interaction 1, the stars' coordinates can be as follows:
- star $$$1$$$ is at $$$(1, 0, 0)$$$,
- star $$$2$$$ is at $$$(-1, -1, 3)$$$, and
- star $$$3$$$ is at $$$(-2, 0, 0)$$$.
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| Figure F.1: Illustration of Sample Interaction 1 |
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In Sample Interaction 2, the stars' coordinates can be as follows:
- star $$$1$$$ is at $$$(-3998, -3998, -3997)$$$,
- star $$$2$$$ is at $$$(3997, 3997, 3996)$$$,
- star $$$3$$$ is at $$$(-3999, -3998, -3998)$$$, and
- star $$$4$$$ is at $$$(3999, 3999, 3998)$$$.
In Sample Interaction 3, the stars' coordinates can be as follows:
- star $$$1$$$ is at $$$(1, 0, 0 )$$$,
- star $$$2$$$ is at $$$(0, -2, 0 )$$$,
- star $$$3$$$ is at $$$(0, 0, 3 )$$$,
- star $$$4$$$ is at $$$(3, -4, 5 )$$$, and
- star $$$5$$$ is at $$$(-6, 7, -8)$$$.
