In the \(xy\) coordinate system, a massive particle is initially at rest at \((x,y) = (\frac{1}{\sqrt{2}}m,2m)\). Under the influence of gravity \((10 \frac{m}{s^{2}}\) in the -y direction\()\), the particle falls and then bounces off of a unit circle situated at the origin. It then continues on through space until its path intersects the \(x\) axis.

How far is the particle (in \(meters\)) from the origin when it intersects the \(x\) axis (to three decimal places)?

**Note:** For our purposes, define a "bounce" as an elastic impact with the surface of an object, such that the normal component of the velocity is reversed and the tangential component is preserved. "Normal" and "tangential" are defined in relation to the surface geometry at the point of impact.

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