Let $C\subset \mathbb{R}^2$ be a smooth, simple closed curve. The thief is inside $C$. Before he starts to move, the police bureau of the $\mathbb{R}^2$ world can freely place countably infinite officers on $C$. We know that

• The thief and the officers move simultaneously and continuously with equal speed.
• The officers are restricted to move on $C$. They can pass right through each other without collision.
• The thief is caught if his coordinates coincide with those of an officer.

Which of the following claim is true?

• The thief always has a plan to get out of $C$.
• The officers always have a plan to prevent the thief from getting out.
• It depends on the shape of $C$.