JMO 1993 Problem 1

Call a word forming by alphabetical small letters $a,b,c,\ \cdots x,y,z$ and a periodic word arranging by a certain word more than two times repeatedly.For example kyonkyon is eight-letter periodic word. Given two words $W_1,\ W_2$ which have the same number of letters and have a different first letter, if you will remove the letter, $W_1$ and $W_2$ will be same word.Prove that either $W_1$ or $W_2$ is not periodic.