
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

Let ellipse $\Gamma: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ have an eccentricity of $\frac{\sqrt{3}}{2}$. A line with slope $k (k > 0)$ passes through the left focus $F$ and intersects the ellipse $\Gamma$ at points $A$ and $B$. If $\overrightarrow{A F}=3 \overrightarrow{F B}$, find $k$.

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