
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 an ellipse \( C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \ (a>b>0) \) that passes through the point \( M(\sqrt{2}, 1) \) and has its left focus at \( F_{1}(-\sqrt{2}, 0) \):

(1) Find the equation of the ellipse \( C \);

(2) When a moving line \( l \) passing through the point \( P(4,1) \) intersects the ellipse \( C \) at two distinct points \( A \) and \( B \), and a point \( Q \) is taken on the segment \( AB \) such that \( |\overrightarrow{A P}| \cdot |\overrightarrow{Q B}| = |\overrightarrow{A Q}| \cdot |\overrightarrow{P B}| \), find the equation of the locus of the point \( Q \).

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