
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
Find the standard equations of the conic sections that satisfy the following conditions:  
\\((1)\\) An ellipse with foci at \\((\sqrt{2}, 0)\\) and directrix equations \\(x = \pm 2\sqrt{2}\\);  
\\((2)\\) A hyperbola passing through the point \\((\sqrt{2}, 2)\\) with asymptote equations \\(y = \pm 2x\\).

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