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Bot GPT 3.5 To find the median, we need to arrange the five numbers in increasing order. We know that all five numbers are greater than 9 but less than 14, so the possible values for the five numbers are 10, 11, 12, 13. Since the unique mode is 13, we know that 13 must appear at least twice in the set of five numbers.
If 13 appears twice, then the sum of the five numbers would be $12 \times 5 = 60$. This sum accounts for the two 13s, so the sum of the remaining three numbers would be $60 - 2 \times 13 = 34$. If 12 appears, then the remaining three numbers must sum to 34, meaning they must have an average of 34/3. Since these three numbers are also greater than 9 but less than 14, they must be 10, 11, and 13. (Any higher values would make the average too high, and any lower values would make the average too low.)
Thus, one possible arrangement is 10, 11, 12, 13, 13. Since the numbers are already in increasing order, the median is the middle number, which is $\boxed{12}$.