
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 point $A(2,0)$, $O$ as the origin, and a moving point $P$ satisfies $|\overrightarrow{OP}+\overrightarrow{OA}|+|\overrightarrow{OP}-\overrightarrow{OA}|=4\sqrt{2}$.

$(1)$ Find the standard equation of the trajectory $C$ of the moving point $P$;

$(2)$ A line $l$ passes through point $A$ and is not perpendicular to the coordinate axes, intersects the trajectory $C$ at two distinct points $M$, $N$. The perpendicular bisector of segment $MN$ intersects the $x$-axis at point $D$, and the midpoint of segment $MN$ is $H$. Find the range of values for $\dfrac{|DH|}{|MN|}$.

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