
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $M(\sqrt{2}, 1)$, and its eccentricity is $\frac{\sqrt{2}}{2}$.

$(1)$ Find the equation of the ellipse $C$.

$(2)$ A line $l$ passing through the point $Q(1, 0)$ intersects the ellipse $C$ at points $A$ and $B$. For point $P(4, 3)$, let the slopes of lines $PA$ and $PB$ be $k_1$ and $k_2$ respectively. When $k_1 \cdot k_2$ is maximized, find the equation of line $l$.

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