Name: SAT Math Help & School Math Mastery
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A sequence of numbers is defined recursively by $a_{1} = 3$ , $a_{2} = 2$ , and

a_{n} = \frac{a _{n - 2} \cdot a _{n - 1}}{2 a _{n - 2} - a _{n - 1}}

for all $n \geq 3$ . Then $a_{2024}$ can be written as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. What is $p + q$ ?