
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 $A(2,0)$ and $O$ is the origin, the moving point $P$ satisfies $|\vec{OP} + \vec{OA}| + |\vec{OP} - \vec{OA}| = 4\sqrt{2}$:
  - (Ⅰ) Find the equation of the trajectory C of the moving point $P$.
  - (Ⅱ) A line $l$, passing through point $A$ and not perpendicular to the coordinate axes, intersects the trajectory $C$ at two distinct points $M$ and $N$. The perpendicular bisector of the segment $MN$ intersects the $x$-axis at point $D$. Let $H$ be the midpoint of the segment $MN$. Determine the range of values for $\frac{|DH|}{|MN|}$.

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