
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
Given a parabola $C$ with its vertex at the origin and its focus $F$ on the positive half of the $x$-axis. A line $l$ passing through point $F$ intersects the parabola $C$ at points $A$ and $B$, satisfying $\overrightarrow{OA} \cdot \overrightarrow{OB} = -\frac{3}{4}$.
$(1)$ Find the standard equation of the parabola $C$;
$(2)$ If point $M$ moves on the directrix of parabola $C$ with its $y$-coordinate ranging from $[-1,1]$, and $\overrightarrow{MA} \cdot \overrightarrow{MB} = 9$, point $N$ is a common point of the circle with diameter $AB$ and the directrix of parabola $C$, find the range of the $y$-coordinate of point $N$.

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