dna reproduction speed

How is it possible for DNA to reproduce at the speeds it does? DNA
bases find their place at a rate of up to 1000 base pairs per second
at a replication site. I have a hard time picturing this.  Imagine a
string encrusted with magnets, thrown into a bag of short magnetic
clusters and shaken rapidly. I can imagine, if the design was clever
enough, that occasionally a cluster would find its proper place at the
end of the string, as the clusters got knocked around and one just
happened to land at the right angle to get knocked into the right
place to join at the end of the string.  But this would be a very slow
process. Most of the time, the clusters would block each other by
fitting in a not quite perfect way (a local energy minimum) rather
than instantly finding a perfect fit (a global energy minimum). How
does it happen so fast in the cell?
If you watch visualization movies of the process, it looks like a time-
reversed video, because it is so clearly going from a disordered state
(base pairs floating nearby) to an ordered state (a DNA molecule) very
quickly.