
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 \( x^{2} - y^{2} = 2 \) with the left and right foci as \( F_{1} \) and \( F_{2} \) respectively, and a moving straight line passing through \( F_{2} \) intersects the hyperbola at points \( A \) and \( B \).

(1) If a moving point \( M \) satisfies \( \overrightarrow{F_{1} M} = \overrightarrow{F_{1} A} + \overrightarrow{F_{1} B} + \overrightarrow{F_{1} O} \) (where \( O \) is the origin), find the equation of the locus of point \( M \).

(2) Is there a fixed point \( C \) on the \( x \)-axis such that \( \overrightarrow{C A} \cdot \overrightarrow{C B} \) is a constant? If it exists, find the coordinates of point \( C \); if it does not exist, explain why.

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