Let $\mathbf{u}$ and $\mathbf{v}$ be vectors such that $\|\mathbf{u}\| = \|\mathbf{v}\| = 2$ and $\mathbf{u} \cdot \mathbf{v} = -1.$  If $\theta$ is the angle between the vectors $\mathbf{u} + \mathbf{v}$ and $2 \mathbf{u} - \mathbf{v},$ then find $\cos \theta.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $\mathbf{u}$ and $\mathbf{v}$ be vectors such that $\|\mathbf{u}\| = \|\mathbf{v}\| = 2$ and $\mathbf{u} \cdot \mathbf{v} = -1.$  If $\theta$ is the angle between the vectors $\mathbf{u} + \mathbf{v}$ and $2 \mathbf{u} - \mathbf{v},$ then find $\cos \theta.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{1}{4}
Let $\mathbf{u}$ and $\mathbf{v}$ be vectors such that $\|\mathbf{u}\| = \|\mathbf{v}\| = 2$ and $\mathbf{u} \cdot \mathbf{v} = -1.$  If $\theta$ is the angle between the vectors $\mathbf{u} + \mathbf{v}$ and $2 \mathbf{u} - \mathbf{v},$ then find $\cos \theta.$
train