
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 system, given point \( A(\frac{1}{2},0) \), vector \( \vec{e} = (0,1) \), and point B moving along the line \( x=-\frac{1}{2} \), find point C such that \( 2\vec{OC}=\vec{OA}+\vec{OB} \), and point M such that \( \vec{BM} \cdot \vec{e} = 0 \) and \( \vec{CM} \cdot \vec{AB} = 0 \).
(1) Determine the equation for the trajectory E of the moving point M.
(2) Let point P be a moving point on trajectory E, with points R and N on the y-axis, and a circle inscribed in triangle PRN with equation \( (x-1)^2 + y^2 = 1 \). Find the minimum value of the area of triangle PRN.

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