Let $\mathbf{a}$ and $\mathbf{b}$ be unit vectors such that $\mathbf{a} + 2 \mathbf{b}$ and $5 \mathbf{a} - 4 \mathbf{b}$ are orthogonal.  Find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees.

Note: A unit vector is a vector of magnitude 1.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $\mathbf{a}$ and $\mathbf{b}$ be unit vectors such that $\mathbf{a} + 2 \mathbf{b}$ and $5 \mathbf{a} - 4 \mathbf{b}$ are orthogonal.  Find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees.

Note: A unit vector is a vector of magnitude 1.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
60^\circ
Let $\mathbf{a}$ and $\mathbf{b}$ be unit vectors such that $\mathbf{a} + 2 \mathbf{b}$ and $5 \mathbf{a} - 4 \mathbf{b}$ are orthogonal.  Find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees.

Note: A unit vector is a vector of magnitude 1.
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