
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 ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passing through the point $M(\sqrt{2}, 1)$ with an eccentricity of $\frac{\sqrt{2}}{2}$.
1. Find the equation of the ellipse $C$.
2. The line $l$ passing through the point $Q(1, 0)$ intersects the ellipse $C$ at points $A$ and $B$. Let $P(4, 3)$ be a point, and denote the slopes of lines $PA$ and $PB$ as $k_1$ and $k_2$, respectively. Find the equation of line $l$ when $k_1 \cdot k_2$ is at its maximum.

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