
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, whose left and right foci are $F_1$ and $F_2$ respectively. A circle centered at $F_1$ with a radius of $3$ intersects with another circle centered at $F_2$ with a radius of $1$, and their intersection points lie on the ellipse $C$. Let point $A(0,b)$ be on $\triangle AF_1F_2$ with $\angle F_1AF_2 = \frac{2\pi}{3}$.
1. Find the equation of the ellipse $C$.
2. Suppose a line $l$ passing through point $P(2,-1)$ does not pass through point $A$ and intersects the ellipse $C$ at points $M$ and $N$. If the slopes of lines $AM$ and $AN$ are $k_1$ and $k_2$ respectively, find the value of $k_1 + k_2$.

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