
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let $O$ be the origin of the coordinates. Given an ellipse $C_{1}: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with eccentricity $\frac{\sqrt{3}}{2}$, and a parabola $C_{2}: x^{2} = -ay$ with directrix $y = \frac{1}{2}$.

$(1)$ Find the equations of the ellipse $C_{1}$ and the parabola $C_{2}$.

$(2)$ Suppose a line $t$ passing through the fixed point $M(0,2)$ intersects the ellipse $C_{1}$ at two distinct points $P$ and $Q$. If point $O$ is outside of the circle with diameter $PQ$, find the range of the slope $k$ of line $t$.

Present the answer in LaTex format: \boxed{Your answer}
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