
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
The hyperbola and the ellipse, both centered at the origin \\(O\\) and symmetric with respect to the coordinate axes, have a common focus. Points \\(M\\) and \\(N\\) are the two vertices of the hyperbola. If \\(M\\), \\(O\\), and \\(N\\) divide the major axis of the ellipse into four equal parts, then the ratio of the eccentricities of the hyperbola and the ellipse is

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