
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
A straight line $l: (m+1)x + 2y - 2m - 2 = 0$ $(m \in \mathbb{R})$ always passes through a fixed point $C$. Taking $C$ as the center and 2 as the radius, a circle $C$ is constructed.  
(1) Find the equation of circle $C$;  
(2) Assume the point $C$ has a symmetrical point $C^{1}$ with respect to the y-axis. A moving point $M$ lies on the curve $E$, and in the triangle $\triangle MCC'$, it satisfies that $\angle C^{1}MC = 2\theta$ and the area of triangle $\triangle MCC'$ is $4\tan\theta$. Find the equation of curve $E$;  
(3) A point $P$ lies on curve $E$ from part (2). Two tangents to circle $C$ are drawn from point $P$, with points of tangency at $Q$ and $R$. Find the minimum value of $\vec{PQ} \cdot \vec{PR}$.

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