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3D compass point from 3D coordinates

Given a set of Cartesian (x, y, z) coordinates, output the 3D compass point that comes nearest to describing its direction from the origin.

In reality, the direction of "North" changes as you move across the surface of the globe. This challenge is based in a simplified universe where the compass points always denote the same direction, but they are in 3D, defined as follows:

- East: the positive x direction, towards (1, 0, 0)
- West: the negative x direction, towards (-1, 0, 0)
- North: the positive y direction, towards (0, 1, 0)
- South: the negative y direction, towards (0, -1, 0)
- Up: the positive z direction, towards (0, 0, 1)
- Down: the negative z direction, towards (0, 0, -1)

A compass point consists of 1, 2, or 3 letters from EWNSUD (being the initial letters of the directions mentioned). A compass point is defined as follows (the definitions in _**bold italics**_ deliberately use different terms than real world compass points to avoid the conflict of redefining existing terms):

- A _**primary compass point**_ is a single letter
- A _**secondary compass point**_ is 2 letters, denoting the direction half way between them.
  - Opposing directions cannot be combined, so EW, NS, and UD are not valid compass points
  - A compass point will never end in a repeated character
  - The order of letters in a secondary compass point are always z axis direction, then y axis direction, then x axis direction. So NE rather than EN, DN rather than ND, and UW rather than WU (the same does **not** apply to a tertiary compass point, where the direction is dependent on the order of the letters)
- A _**tertiary compass point**_ is 3 letters, which is interpreted as a primary compass point (1 letter) followed by a secondary compass point (2 letters), denoting the direction half way between them
  - The modifying primary compass point is either one of the 2 directions in the secondary compass point, or perpendicular to both. For example, NW can be modified by N or W, to give NNW or WNW, or by U or D (perpendicular to both N and W), to give UNW or DNW. NW **cannot** be modified by E or S (this would give additional names to already named directions). So ENW and SNW are not valid (ENW is exactly equivalent to the valid NNE and SNW is exactly equivalent to the valid WSW)

## Input
- The input will be 3 coordinates in the order x, y, z
- You may choose to take these coordinates in any format that does not already provide part of the solution
- Each of x, y, and z will be in the range [-1, 1] where square brackets indicate that the endpoints are inclusive
- The input will never be (0, 0, 0)

## Output
- A string containing 1, 2, or 3 uppercase letters as described above
- If more than 1 compass point is nearest to the direction of the input from the origin, any of them is a valid output
- The output must be one of the valid compass points defined above. Each one uniquely defines a direction - there are no 2 compass points that define the same direction. The complete list follows:
  - 6 primary compass points: E, W, N, S, U, D
  - 12 secondary compass points: UN, US, UE, UW, DN, DS, DE, DW, NE, NW, SE, SW
  - 48 tertiary compass points: UUN, NUN, EUN, WUN, UUS, SUS, EUS, WUS, UUE, EUE, NUE, SUE, UUW, WUW, NUW, SUW, DDN, NDN, EDN, WDN, DDS, SDS, EDS, WDS, DDE, EDE, NDE, SDE, DDW, WDW, NDW, SDW, NNE, ENE, UNE, DNE, NNW, WNW, UNW, DNW, SSE, ESE, USE, DSE, SSW, WSW, USW, DSW  
  - In short:
    - a single letter
    - or two letters that are neither identical nor opposing
    - or three letters that end in a valid two letter combination and contain no two opposing letters (never containing both E and W, or both N and S, or both U and D)

## Test cases
Test cases are in the format `x, y, z : output`

```text
0, 0, 1 : U
0, 0, -1 : D
0, 1, 0 : N
0, -1, 0 : S
1, 0, 0 : E
-1, 0, 0 : W
0, 0.7071, 0.7071 : UN
0, -0.7071, 0.7071 : US
0.7071, 0, 0.7071 : UE
-0.7071, 0, 0.7071 : UW
0, 0.7071, -0.7071 : DN
0, -0.7071, -0.7071 : DS
0.7071, 0, -0.7071 : DE
-0.7071, 0, -0.7071 : DW
0.7071, 0.7071, 0 : NE
-0.7071, 0.7071, 0 : NW
0.7071, -0.7071, 0 : SE
-0.7071, -0.7071, 0 : SW
0, 0.3827, 0.9239 : UUN
0, -0.3827, 0.9239 : UUS
0.3827, 0, 0.9239 : UUE
-0.3827, 0, 0.9239 : UUW
0.5, 0.5, 0.7071 : UNE
-0.5, 0.5, 0.7071 : UNW
0.5, -0.5, 0.7071 : USE
-0.5, -0.5, 0.7071 : USW
0, 0.3827, -0.9239 : DDN
0, -0.3827, -0.9239 : DDS
0.3827, 0, -0.9239 : DDE
-0.3827, 0, -0.9239 : DDW
0.5, 0.5, -0.7071 : DNE
-0.5, 0.5, -0.7071 : DNW
0.5, -0.5, -0.7071 : DSE
-0.5, -0.5, -0.7071 : DSW
0, 0.9239, 0.3827 : NUN
0.5, 0.7071, 0.5 : NUE
-0.5, 0.7071, 0.5 : NUW
0, 0.9239, -0.3827 : NDN
0.5, 0.7071, -0.5 : NDE
-0.5, 0.7071, -0.5 : NDW
0.3827, 0.9239, 0 : NNE
-0.3827, 0.9239, 0 : NNW
0, -0.9239, 0.3827 : SUS
0.5, -0.7071, 0.5 : SUE
-0.5, -0.7071, 0.5 : SUW
0, -0.9239, -0.3827 : SDS
0.5, -0.7071, -0.5 : SDE
-0.5, -0.7071, -0.5 : SDW
0.3827, -0.9239, 0 : SSE
-0.3827, -0.9239, 0 : SSW
0.7071, -0.5, 0.5 : EUS
0.7071, 0.5, 0.5 : EUN
0.9239, 0, 0.3827 : EUE
0.7071, 0.5, -0.5 : EDN
0.7071, -0.5, -0.5 : EDS
0.9239, 0, -0.3827 : EDE
0.9239, 0.3827, 0 : ENE
0.9239, -0.3827, 0 : ESE
-0.7071, 0.5, 0.5 : WUN
-0.7071, -0.5, 0.5 : WUS
-0.9239, 0, 0.3827 : WUW
-0.7071, 0.5, -0.5 : WDN
-0.7071, -0.5, -0.5 : WDS
-0.9239, 0, -0.3827 : WDW
-0.9239, 0.3827, 0 : WNW
-0.9239, -0.3827, 0 : WSW
```

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# Sandbox questions
- Is it more useful to include or omit parentheses in the test case format?
- Are there any edge cases that should be included in the test cases?


## TO DO BEFORE POSTING
- ADD DIAGRAMS SHOWING THE PRIMARY, SECONDARY, AND TERTIARY COMPASS POINTS. MAYBE ONE FOR PRIMARY, ONE FOR SECONDARY, ONE FOR TERTIARY, AND ONE SHOWING THEM ALL TOGETHER. MAYBE COMMENT ON WHETHER THEY ARE EQUALLY SPACED OVER THE SPHERE
- HAVE NO TEST CASES THAT HAVE MORE THAN 1 VALID OUTPUT, BUT INCLUDE CASES VERY CLOSE TO THOSE POINTS - INCLUDE CASES CLOSE TO EQUIDISTANT FROM 2 DIRECTIONS (A CASE FROM EACH SIDE) AND CASES CLOSE TO EQUIDISTANT FROM 3 DIRECTIONS (A CASE FROM ALL 3 SIDES, POTENTIALLY 6 TO COVER BOTH OPTIONS FOR SECOND CLOSEST TOO)
- ENSURE THE TEST CASE OUTPUTS INCLUDE ALL POSSIBLE COMPASS POINTS (THERE ARE FEW ENOUGH TO MAKE THIS FEASIBLE)