#Pyth, 33 bytes#

This is my second attempt on Pyth :)

KJ<z/t@+1*8lz2 2;WnkJ=K+K=JtJ;qKz

Try it online!

How it works:

KJ<z/t@+1*8lz2 2;WnkJ=K+K=JtJ;qKz
                                    z=input()
KJ<z/t@+1*8lz2 2                    K=J=z[:(root(1+8*len(z),2)-1)/2]
  <z                                    z[:
       +1*8lz                                    1+8*len(z)
      @      2                              root(          ,2)
     t                                                        -1
                 WnKJ               while not_equal(k,J):
                     =K+K=JtJ        K=plus(K,J=tail(J))
                              qKz   equal(K,z)