A particle moves along the \(x\)-axis with an initial velocity \(v_0 > 0\) at position \(x = 0\) when time \(t = 0\). The acceleration of the particle depends on its instantaneous velocity according to the relation \(a(v) = -kv\), where \(k\) is a positive constant with units of \(\text{s}^{-1}\). Which of the following statements correctly describes the speed of the particle and its total distance traveled in the limit \(t \to \infty\)?