Equilateral triangle $OAB$ has side length 1.  The point $P$ satisfies
\[\overrightarrow{OP} = (2 - t) \overrightarrow{OA} + t \overrightarrow{OB},\]where $t$ is a real number.  Find the minimum value of $|\overrightarrow{AP}|.$
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Equilateral triangle $OAB$ has side length 1.  The point $P$ satisfies
\[\overrightarrow{OP} = (2 - t) \overrightarrow{OA} + t \overrightarrow{OB},\]where $t$ is a real number.  Find the minimum value of $|\overrightarrow{AP}|.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
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user
\frac{\sqrt{3}}{2}
Equilateral triangle $OAB$ has side length 1.  The point $P$ satisfies
\[\overrightarrow{OP} = (2 - t) \overrightarrow{OA} + t \overrightarrow{OB},\]where $t$ is a real number.  Find the minimum value of $|\overrightarrow{AP}|.$
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