
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 moving point $P$ whose distance to a fixed point $F(\sqrt{2},0)$ and its distance to a fixed line $l: x=2\sqrt{2}$ have a ratio of $\frac{\sqrt{2}}{2}$.
$(1)$ Find the equation of the trajectory $C$ of the moving point $P$;
$(2)$ Let $M$ and $N$ be two points on the line $l$, and point $E$ is symmetric to point $F$ with respect to the origin $O$. If $\overrightarrow{EM} \cdot \overrightarrow{FN} = 0$, find the minimum value of $|MN|$.

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