
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 that \( F_{1} \) and \( F_{2} \) are the foci of the hyperbola \( x^{2} - y^{2} = 1 \), \( M \) is a point on the right branch of the hyperbola, and \( O \) is the origin of the coordinate system. If \( \frac{\left|M F_{1}\right| + \left|M F_{2}\right|}{|M O|} = \sqrt{6} \), then the coordinates of point \( M \) are ____.

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