MathNet 2021 — Part 02 Question 67

Suppose $m$ is a real number, and complex numbers $z_1 = 1 + 2i$, $z_2 = m + 3i$, where $i$ is the imaginary unit. If $z_1 \cdot \bar{z_2}$ is purely imaginary, then the value of $|z_1 + z_2|$ is ______.