BALKAN-SL 2018 C3

An open necklace can contain rubies, emeralds and sapphires. At every step we can perform any of the following operations: $\left(1^{\circ}\right)$ We can replace two consecutive rubies with an emerald and a sapphire, where the emerald is on the left of the sapphire. $\left(2^{\circ}\right)$ We can replace three consecutive emeralds with a sapphire and a ruby, where the sapphire is on the left of the ruby. (3 ) If we find two consecutive sapphires then we can remove them. (4 ) If we find consecutively and in this order a ruby, an emerald, and a sapphire, then we can remove them.

Furthermore we can also reverse all of the above operations. For example, by reversing $\left(3^{\circ}\right)$ we can put two consecutive sapphires on any position we wish. Initially the necklace has one sapphire (and no other precious stones). Decide, with proof, whether there is a finite sequence of steps such that at the end of this sequence the necklace contains one emerald (and no other precious stones). Remark. A necklace is open if its precious stones are on a line from left to right. We are not allowed to move a precious stone from the rightmost position to the leftmost as we would be able to do if the necklace was closed. (Cyprus)