
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
An ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (with $a > b > 0$) passes through the point $A(1,\frac{3}{2})$ and has an eccentricity of $\frac{1}{2}$. The left and right foci are located in the range $\{x | -2 \leq x < 0 \}$. A line passing through focus $F_1$ intersects the ellipse at points $A$ and $B$.

(Ⅰ) Find the equation of ellipse $C$.

(Ⅱ) When the area of triangle $\Delta F_2AB$ is $\frac{12\sqrt{2}}{7}$, find the equation of the line.

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