
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 an ellipse $C$ with one focus at $F_{1}(2,0)$ and the corresponding directrix $x=8$, and eccentricity $e=\frac{1}{2}$.

$(1)$ Find the equation of the ellipse $C$;

$(2)$ Find the length of the chord cut from the ellipse $C$ by a line passing through the other focus and having a slope of $45^{\circ}$.

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