
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 \( C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) where \( a > 0 \) and \( b > 0 \) has an eccentricity of 2, a line \( l \) with a slope of 1 passes through the point \( P(0, m) \) where \( m > 0 \). The line intersects the hyperbola \( C \) at points \( A \) and \( B \) such that
\[ \overrightarrow{AP} = 3 \overrightarrow{PB} \]
and
\[ \overrightarrow{OA} \cdot \overrightarrow{OB} = 3. \]
(1) Find the equation of the hyperbola.

(2) Let \( Q \) be a moving point on the hyperbola \( C \), and let \( F \) be the left focus of the hyperbola \( C \). Is there a fixed point \( M \) on the negative half-axis of the \( x \)-axis such that \( \angle QFM = 2 \angle QMF \)? If such a point exists, find the coordinates of \( M \); if not, explain the reason.

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