
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 the ellipse $C: 2x^{2}+y^{2}=16$.
$(1)$ Find the eccentricity of ellipse $C$;
$(2)$ Let $O$ be the origin, point $A$ on ellipse $C$, and point $B$ on the line $x=4$, and $\overrightarrow{OA} \cdot \overrightarrow{OB}=0$, find the length $l$ of the chord cut from the circle $x^{2}+y^{2}=17$ by line $AB$.

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