
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 that point $D$ is a point on the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and $F_1, F_2$ are the left and right foci of $C$ respectively, with $|F_1F_2| = 2\sqrt{2}, \angle F_1DF_2 = 60^{\circ}$, and the area of $\Delta F_1DF_2$ is $\frac{2\sqrt{3}}{3}$.

1. Find the equation of the ellipse $C$;
2. Line $l$ passes through point $Q(1, 0)$ and intersects ellipse $C$ at points $A$ and $B$. Point $P(4, 3)$ is given, and the slopes of lines $PA$ and $PB$ are $k_1$ 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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