A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ such that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$. If $\omega_1$ has a diameter of 1 and touches the parabola at $(0,0)$, find the diameter of $\omega_{1998}$.

Let's think step by step and output the final answer within \boxed{}.
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