
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
In the ellipse $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1\; (a > b > 0)$, a line with slope $k(k > 0)$ intersects the ellipse at the left vertex $A$ and another point $B$. The projection of point $B$ on the $x$-axis is exactly the right focus $F$. If the eccentricity of the ellipse $e= \frac{1}{3}$, then the value of $k$ is _____.

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