
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 $F\_1$ and $F\_2$ are the left and right foci of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and the sum of the distances from point $A(0, \sqrt{3})$ on the ellipse $C$ to $F\_1$ and $F\_2$ is $4$,
(1) Find the equation of the ellipse $C$;
(2) Find the length of the minor axis and the focal length of the ellipse $C$.

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