The velocity of a particle for $t\geq0$ is given by the vector equation $\vec{v}=\left(2+2\sin{2t}\right)\vec{i}+\left(2+t\cos{2t}\right)\vec{j}$.

For the interval $\left[0, \cfrac{\pi}{2}\right]$ what is the maximum velocity of the particle in the horizontal direction?

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