The following is a question in an entrance examination of a Japanese university.

A quadrilateral $ABCD$ is inscribed in a circle with a radius of $65/8$. The perimeter of this quadrilateral is $44$ and the lengths of $BC$ and $CD$ are both $13$. What are the lengths of the remaining two sides $AB$ and $DA$?

The answer is $AB = 14, DA = 4$ or $AB = 4, DA = 14$.

I think if such a quadrilateral did not exist, this problem would be a very bad problem. So I think we must check the existence of such a quadrilateral. I asked a man about my question. He said $A \implies B$ is true even if $A$ is not true.

Which is correct, me or him?