3. In triangle $A B C$, whose incenter is $I$, it holds that

$$
|C A|+|A I|=|B C|
$$

Find the ratio of the measures of angles $\angle B A C$ and $\angle C B A$.
Let's choose a point $D, D \notin AC$, on the semicircle $CA$, such that $|AD|=|AI|$. Then, $DBC$ is an isosceles triangle with vertex $C$, and since $I$ lies on the angle bisector of $\angle DCB$, we have $\angle IDC = \angle CBI$. Since $ADI$ is an isosceles triangle with vertex $A$, we have $\angle IAC = 2 \angle IDC$. Given that $\angle IAC = \frac{1}{2} \angle BAC$ and $\angle CBI = \frac{1}{2} \angle CBA$, it follows that $\angle BAC = 2 \angle CBA$ or $\angle BAC : \angle CBA = 1 : 2$.

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