
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 ellipse $C_1: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) passes through the point $(1, e)$, where $e$ is the eccentricity of the ellipse $C_1$. A circle $C_2$ with the center at the origin $O$ and the diameter equal to the major axis length of the ellipse $C_1$ is tangent to the line $x - y + 2 = 0$.

(Ⅰ) Find the equations of the ellipse $C_1$ and the circle $C_2$;  
(Ⅱ) A line $l_1$ passing through the right focus $F$ of the ellipse $C_1$ intersects the ellipse at points $A$ and $B$. A line $l_2$ passing through $F$ and perpendicular to $l_1$ intersects the circle $C_2$ at points $C$ and $D$. Let the quadrilateral with vertices $A$, $B$, $C$, $D$ have an area denoted as $S$. Find the range of possible values for $S$.

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