
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
In the Cartesian coordinate plane, point $A(\frac{1}{2}, 0)$ is given. Point $B$ is a moving point on the line $x = -\frac{1}{2}$, and point $C$ is the intersection of line segment $AB$ and the $y$-axis. Point $M$ satisfies $\overrightarrow{BM} \cdot \overrightarrow{OC} = 0$ and $\overrightarrow{CM} \cdot \overrightarrow{AB} = 0$.
(1) Find the equation of the trajectory $E$ of moving point $M$;
(2) Assume point $P$ is a moving point on trajectory $E$, and points $R$ and $N$ are on the $y$-axis. The circle $(x - 1)^2 + y^2 = 1$ is internally tangent to $\triangle PRN$. Find the minimum value of the area of $\triangle PRN$.

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