Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be three vectors with equal magnitudes, and such that any pair of them are orthogonal.  If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{a} + \mathbf{b} + \mathbf{c},$ then find $\cos \theta.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be three vectors with equal magnitudes, and such that any pair of them are orthogonal.  If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{a} + \mathbf{b} + \mathbf{c},$ then find $\cos \theta.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{1}{\sqrt{3}}
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c}$ be three vectors with equal magnitudes, and such that any pair of them are orthogonal.  If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{a} + \mathbf{b} + \mathbf{c},$ then find $\cos \theta.$
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