
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 the focus of the parabola $y^{2}=4x$ is $F$, and $O$ is the origin. $M\left(4,0\right)$ is a point. The line passing through point $F$ intersects the parabola at points $A$ and $B$. The extensions of $AM$ and $BM$ intersect the parabola at points $C$ and $D$ respectively. $P$ and $Q$ are the midpoints of $AB$ and $CD$ respectively.  
$(1)$ Find the range of slopes of the line $OP$.  
$(2)$ Find the minimum value of $\cos \angle POQ$.

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