
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
Let $F_1$ and $F_2$ be the left and right foci of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), respectively. A line $l$ passing through $F_2$ intersects the ellipse $C$ at points $A$ and $B$. The slope angle of line $l$ is $60^\circ$, and the distance from $F_1$ to line $l$ is $2$.
(1) Find the focal distance of the ellipse $C$;
(2) If $b = 2$, find the equation of the ellipse $C$.

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