How to solve for coordinates given their distances

In the game "No Man's Sky Expedition 14", these are the jump distances from the starting system R0 to the five rendezvous systems R1 thru R5. If R0 is (0,0,0) and R1 is (105,0,0) and R4 is in the x-y plane, what are the coordinates for R2 thru R5?

  R0 -> R1 105 ly
  R0 -> R2 259 ly
  R0 -> R3 360 ly
  R0 -> R4  33 ly
  R0 -> R5 235 ly
  R1 -> R2 255 ly
  R1 -> R3 373 ly
  R1 -> R4  76 ly
  R1 -> R5 187 ly
  R2 -> R3 251 ly
  R2 -> R4 269 ly
  R2 -> R5 201 ly
  R3 -> R4 370 ly
  R3 -> R5 229 ly
  R4 -> R5 225 ly

Turns out that I can't say that P4 is in the x-y plane because that makes P2 unsolvable. So I don't know how to approach this. Any help appreciated.

Take any three systems

R0 -> R1 : 105
R0 -> R2 : 259
R1 -> R2 : 255

that form the three sides of a triangle.

Then use the cosine rule to find the angles between each of two sides

and then use the distances and angles to find (x,y) coordinates. This is just a polar to cartesian conversion.