A Confusing Logic Puzzle

Logic Level 5

Jane, Emily, and Mike are perfect logicians. One day, Jane said, "I'm thinking of four non-negative integers a,b,x,a, b, x, and yy that obey the following conditions: ax1axmin(b,y)ax1+min(b,y)by1.\begin{aligned} |a - x| &\geq 1\\ |a - x| &\geq \min(b, y)\\ |a - x| &\leq 1 + \min(b, y)\\ |b - y| &\leq 1. \end{aligned} Then Jane said, "I'm going to tell aa and bb to Emily and xx and yy to Mike."

Emily said, "I don't know xx, and I wouldn't know it even if I knew whether yy was the same as bb."
Mike said, "I don't know aa, and I wouldn't know it even if I knew whether bb was the same as yy."
Emily said, "I don't know xx, and I wouldn't know it even if I knew whether yy was the same as bb."
Mike said, "I don't know aa, and I wouldn't know it even if I knew whether bb was the same as yy."
Emily said, "I don't know xx, and I wouldn't know it even if I knew whether yy was the same as bb."
Mike said, "I don't know aa, and I wouldn't know it even if I knew whether bb was the same as yy."

Jane then interrupted, "Stop! You two could go on forever like that!"

Emily said, "I didn't know that."
Mike said, "I didn't know Emily didn't know that. If Emily had said she knew that, I wouldn't know whether Emily knew whether xx is greater than 1515. But now, I do."
Emily said, "Before Mike said that, I didn't know whether Mike knew which of aa and xx is bigger."

What is the maximum value of a×b+x×y?a \times b+x \times y?

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