SPECIAL TRAPEZIUM (Difficult)

Alice likes $isosceles$ $trapezium$ and is fond of drawing different types of shapes using it. She thought of making $a$ $circular$ $closed$ $loop$ by joining the edges of isosceles trapezium. She starts joining isosceles trapeziums but she is curious about knowing the perimeter $P$ of that circular closed loop before drawing it physically. Among her many friends she finds you noble and challen