
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 $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a > b > 0\right)$, where the product of the slopes of the lines connecting the upper vertex of the ellipse with its left and right vertices is $-\frac{1}{4}$. <br/>$(1)$ Find the eccentricity of the ellipse $C$; <br/>$(2)$ If the line $y=\frac{1}{2}(x+1)$ intersects the ellipse $C$ at points $A$ and $B$ such that $|AB|=\frac{\sqrt{35}}{2}$, find the standard equation of the ellipse $C$.

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