There is a rectangular field that measures $20\,\text{m}$ by $15\,\text{m}$. Xiaoyu the butterfly is sitting at the perimeter of the field on one of the $20\,\text{m}$ sides such that he is $6\,\text{m}$ from a corner. He flies in a straight line to another point on the perimeter. His flying path splits the field into two parts with equal area. How far in meters did Xiaoyu fly?

Let's think step by step and output the final answer within \boxed{}.
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