When the winning strategy receives piles on which she can perform multiple operations in order to achieve bitwise_XOR=0….0, how do we know which operation it performs?

For instance, if on it's turn it receives the piles (in binary rep): A=0100 B=0100 and C=1000,

C can be reduced to either 0100 or 0000, and the bitwise_XOR will be 0000.

I assume that in this case it would reduce C to 0000, to avoid giving the other strategy the opportunity to maintain the bitwise_XOR=0000 at the end of it's turn. This is not explicitly specified in the winning strategy description however, and unfortunately I am unable to understand it's code.

This affects the answer to the bonus-of-bonus question, so a clarification would be nice.

Thankyou