How to imagine a classical phase space for N particles?

Classically a single particle will have 3 position coordinates and 3 momentum coordinates, and so it "exists" in a 6-dimensional phase space and moves around this space in relation to time (known as the phase trajectory). However I've read that when we have N classical particles, their position/momentum coordinates represent a 6N-dimensional space.

Is this really a 6N-dimensional space? Or is it a 6 dimensional space (representing the possible momentum/position values possible) with N particles defined within that space each with their own trajectories around it?


I guess what I mean, is it one of these two situations;

1) Consider two particles and ignore their momentum for now, only focusing on their x-coordinates of position. Defined by x_1 and x_2 for particles 1 and 2 respectively. Is the phase space here one-dimensional but defined for two particles? As in, do I simply have a co-ordinate axis "x" defined from minus infinity to infinity (a line) and x_1 and x_2 are placed on that coordinate axis appropriately? So if x_1 = 2, x_2 = 100, I place x_1 on 2 and x_2 on 100 on the "x" axis?


or

2) The phase space is actually 2 dimensional and with "x_1" and "x_2" coordinate axes. Like an x-y plane. I place x_1 at (2,0) and x_2 at (0,100)? The value for x_1 can only move along the x_1 coordinate axis, and the value for x_2 can only move along the x_2 coordinate axis.