
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 that point $F(2,0)$ is the right focus of the ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>0,b>0)$, a line passing through point $F$ intersects the ellipse at points $A$ and $B$. If the midpoint of $AB$ has coordinates $({1,-\frac{1}{2}})$, then the eccentricity of $C$ is ____.

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