
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 respectively 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) Suppose points $A$ and $B$ are respectively on the lines $x = -2$ and $x = 2$, and $AF_1 \perpendicular BF_1$.
 (ⅰ) When $\triangle ABF_1$ is an isosceles triangle, find the area of $\triangle ABF_1$;
 (ⅱ) Find the minimum sum of the distances from points $F_1$ and $F_2$ to line $AB$.

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