
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 that point $C$ is the center of the circle $(x+1)^{2}+y^{2}=8$, and $P$ is a moving point on the circle. Point $Q$ lies on the radius $CP$, and there are point $A(1,0)$ and point $M$ on $AP$, satisfying $\overrightarrow{MQ} \cdot \overrightarrow{AP}=0, \overrightarrow{AP}=2 \overrightarrow{AM}$.

$(1)$ When point $P$ moves on the circle, find the equation of the trajectory of point $Q$;

$(2)$ If a line $l$ with slope $k$ is tangent to the circle $x^{2}+y^{2}=1$ and intersects the trajectory of point $Q$ obtained in $(1)$ at two distinct points $F$ and $H$, with $O$ being the origin, and $\frac{3}{4} \leqslant \overrightarrow{OF} \cdot \overrightarrow{OH} \leqslant \frac{4}{5}$, find the range of $k$.

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