
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
1. Given that the eccentricity of an ellipse is $\frac{\sqrt{2}}{2}$ and its latus rectum equation is $x = \pm 8$, find the standard equation of the ellipse.
2. Find the equation of the hyperbola that has common asymptotes with the hyperbola $x^{2} - 2y^{2} = 2$ and passes through the point $M(2, -2)$.

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