
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

Given the points in the plane \( P_{0}(0,0) \), \( P_{1}(1,0) \), and \( P_{2}(1 / 2, \sqrt{3} / 2) \) which are all at a unit distance from each other, determine the coordinates of the point \( P_{3} \) in 3D space such that it is at a distance of 1 from each of the points \( P_{0} \), \( P_{1} \), and \( P_{2} \). Assume the third axis is perpendicular to the plane of the first two axes at \( P_{0} \). No coordinate of \( P_{3} \) should be negative.

In general, provide a method to determine the points \( R_{0}, R_{1}, R_{2}, \ldots, R_{n} \) in \( n \)-dimensional space such that the first \( k \) coordinates of \( R_{k} \) are positive, the remaining coordinates are 0, and the distance between any two points is 1. The distance between points is defined as:

\[
\sqrt{\left(x_{1}-y_{1}\right)^{2}+\left(x_{2}-y_{2}\right)^{2}+\ldots+\left(x_{n}-y_{n}\right)^{2}}
\]

Present the answer in LaTex format: \boxed{Your answer}
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