50-50 chance

An infinite string $a_1 a_2 a_3 \ldots$ of $x$ 's and $y$ 's is constructed as follows:

Step 0. $a_1 =x.$

Step $n, (n > 0).$ If $a_n = x,$ append $xy$ to the end of the sequence. If $a_n = y,$ append $yx$ to the end of the sequence.

What is $a_{1000}?$

The next few steps are shown here for reference.

Step 1. Since $a_1 = x,$ the string $xy$ is appended to the end so the sequence becomes $xxy.$

Step 2. Since $a_2 = x,$ the string $xy$ is appended to the end, so the sequence becomes $xxyxy.$

Step 3. Since $a_3 = y,$ the string $yx$ is appended to the end, so the sequence becomes $xxyxyyx.$