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

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