For $$t \geq 0$$, the components of the velocity of a particle moving in the $$xy$$-plane are given by the parametric equations $$x'(t) = \displaystyle\frac{1}{t+1}$$ and $$y'(t) = ke^{kt}$$, where $$k$$ is a positive constant. The line $$y = 4x + 3$$ is parallel to the line tangent to the path of the particle at the point where $$t = 2$$. What is the value of $$k$$?