
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 \( F_{1} \) and \( F_{2} \) are the left and right foci of the ellipse \( C: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1 \), and points \( A\left(x_{1}, y_{1}\right) \) and \( B\left(x_{2}, y_{2}\right) \) lie on the ellipse \( C \). If \( x_{1}+x_{2}=\frac{1}{2} \) and \( \overrightarrow{A F_{2}}=\lambda \overrightarrow{F_{2} B} \), find the value of \( \lambda \).

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