
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 the ellipse $C_1$: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ ($a>b>0$) has a focal length of 4, with the left and right foci being $F_1$ and $F_2$, respectively. Also, $C_1$ intersects with the parabola $C_2$: $y^2=x$ at points on a line that passes through $F_2$.  
(Ⅰ) Find the equation of the ellipse $C_1$;  
(Ⅱ) Draw parallel lines $m$ and $n$ through $F_1$ and $F_2$, respectively. If line $m$ intersects $C_1$ at points $A$ and $B$, and does not intersect with parabola $C_2$, and line $n$ intersects $C_1$ at points $C$ and $D$, where points $A$ and $D$ are above the x-axis, find the range of the area of quadrilateral $AF_1F_2D$.

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