Thea has a key on her calculator marked X. If an integer is displayed, pressing the X key chops off the first digit and moves it to the end. For example, if $6138$ is on the screen, then pressing theX key changes the display to $1386$. Thea enters a positive integer into her calculator, then squares it, then presses the X key, then squares the result, then presses the X key again. After all these steps, the calculator displays $243$. What number did Thea originally enter?