KöMaL 2023 C1789

A tortoise travelled a distance of $14$ units in the plane. Each of its steps were of a unit's length. After each step it made a turn: if the number of the previous step was odd, then with an angle of $60^{\circ}$, if even, then with an angle of $90^{\circ}$. In steps $3$, $5$, $8$ and $12$ he turned to the right, in the remaining steps he turned to the left. $a)$ Prove that the tortoise will end up in its initial position facing in its initial direction after its $14^{\text{th}}$ step. $b)$ Find the area enclosed by the walk of tortoise in algebraic form.