A particle is at the origin of the Cartesian plane, $(0, 0)$. In each step, the particle can move $1$ unit in any of the four directions: left, right, up, or down. Determine the number of ways the particle can move from $(0, 0)$ to $(0, 2)$ in exactly $6$ steps. 

Note: Two paths are considered different if the order of traversal of each point differs, even if the same set of points is traversed.

Let's think step by step and output the final answer within \boxed{}.
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