Consider the sequence $t_1 = 1$, $t_2 = -1$, and

$$t_n = \left( \frac{n - 3}{n - 1} \right)t_{n-2} \quad \text{for } n \geq 3.$$

Then, the value of the sum

$$\frac{1}{t_2} + \frac{1}{t_4} + \frac{1}{t_6} + \cdots + \frac{1}{t_{2024}}$$

is: