
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 the ellipse $\Gamma$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with its right focus $F(3, 0)$, and upper and lower vertices $A$ and $B$, respectively. The line $AF$ intersects the ellipse $\Gamma$ at another point $M$. If the line $BM$ intersects the $x$-axis at the point $N(12, 0)$, find the eccentricity of $\Gamma$.

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