
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 hyperbola with the equation \(\frac{x^{2}}{16}-\frac{y^{2}}{9}=1\), find a point \(M\) such that \(M F_{1}=2 M F_{2}\). Additionally, derive the equations of the asymptotes and directrices of the hyperbola, and find its eccentricity. Provide a graphical representation.

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