What is the best way to find the period of a (repeating) list in Mathematica?

Question

What is the best way to find the period in a repeating list?

For example:

a = {4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2}

has repeat

Answer 1

Ok, just to show my own work here:

ModifiedTortoiseHare[a_List] := Module[{counter, tortoise, hare},
Quiet[
 Check[
  counter = 1;
  tortoise = a[[counter]];
  hare = a[[2 counter]];
  While[(tortoise != hare) || (a[[counter ;; 2 counter - 1]] != a[[2 counter ;; 3 counter - 1]]),
   counter++;
   tortoise = a[[counter]];
   hare = a[[2 counter]];
  ];
 counter,
$Failed]]]

I'm not sure this is a 100% correct, especially with cases like {pattern,pattern,different,pattern, pattern} and it gets slower and slower when there are a lot of repeating elements, like so:

{ 1,2,1,1, 1,2,1,1, 1,2,1,1, ...} 

because it is making too many expensive comparisons.