
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 its center at the origin, its right focus is at $F(1,0)$, and its eccentricity is $\frac{1}{2}$.
$(1)$ Find the standard equation of ellipse $C$;
$(2)$ A line passing through the right focus of the ellipse and making a $45^\circ$ angle with the x-axis intersects the ellipse at points $A$ and $B$. Find the length of $AB$.

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