
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 an ellipse $C_1$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with a major axis length of 4 and an eccentricity of $\frac{1}{2}$, where $F_1$ and $F_2$ are its left and right foci, respectively. A moving circle passes through point $F_2$ and is tangent to the line $x = -1$.
(Ⅰ) (i) Find the equation of the ellipse $C_1$; (ii) Find the equation of the trajectory of the center $C$ of the moving circle;
(Ⅱ) On the curve $C$, there are two points $M$ and $N$, and on the ellipse $C_1$, there are two points $P$ and $Q$, satisfying that $MF_2$ and $\overrightarrow{NF_2}$ are collinear, $\overrightarrow{PF_2}$ and $\overrightarrow{QF_2}$ are collinear, and $\overrightarrow{PF_2} \cdot \overrightarrow{MF_2} = 0$, find the minimum value of the area of quadrilateral $PMQN$.

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