"XOR Merging" is a game created by Ethen. In the game, the player is given two 01-strings $$$S$$$ and $$$T$$$. The player can "xor merge" any two consecutive digits of $$$S$$$ multiple times, until $$$S$$$ becomes the target string $$$T$$$. When the player "xor merges" the two digits $$$S_i$$$ and $$$S_{i+1}$$$, they will be replaced by the result of $$$S_i$$$ xor $$$S_{i+1}$$$.

For example, let $$$S$$$ be $$$100010111$$$ and $$$T$$$ be $$$101010$$$, the player can convert $$$S$$$ to $$$T$$$ using three "xor merges".

1. First merge $$$S_2=0$$$ and $$$S_3=0$$$ into $$$0$$$, so that $$$S$$$ becomes $$$10010111$$$.

2. Then merge $$$S_3=0$$$ and $$$S_4=1$$$ into $$$1$$$, so that $$$S$$$ becomes $$$1010111$$$.

3. At last merge $$$S_6=1$$$ and $$$S_7=1$$$ into $$$0$$$, so that $$$S$$$ becomes $$$101010$$$, which is the same as $$$T$$$.

However, Ethen found that it is not always possible to convert a given $$$S$$$ to a given $$$T$$$ by "xor merging". Can you help Ethen determine if a solution exists?