Find minimum possible value of each $$$A_i$$$, then check if it gives you $$$B$$$

Can you decrease maximum by half in one operation?

The answer is always <= 4.

If $$$u$$$ and $$$v$$$ both have same remainder when divided by 4, then $$$u \oplus v$$$ is a multiple of $$$4$$$.

XOR Convolution

What should be alice strategy?

Alice should always chose color 1 and 2.

When can Bob color all the vertices using 2 colors?

Bob can color all the vertices using just 2 colors if graph is Bipartite graph

Play as Bob if graph is Bipartite graph and Alice otherwise.

The answer is always YES, if $$$R-L > C$$$ for some $$$C$$$