
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 $F\_1(-1,0)$ and $F\_2(1,0)$ are the left and right foci of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{3} = 1 (a > 0)$,
(I) Find the equation of the ellipse $C$;
(II) If $A$ and $B$ are located on the lines $x = -2$ and $x = 2$ respectively, and $AF\_1 \perp BF\_1$.
(i) When $\triangle ABF\_1$ is an isosceles triangle, find the area of $\triangle ABF\_1$;
(ii) Find the minimum value of the sum of distances from points $F\_1$ and $F\_2$ to line $AB$.

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