
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 $\triangle ABC$, $AB=2$, and $\sin A (1-2\cos B) + \sin B (1-2\cos A) = 0$. Establish a Cartesian coordinate system with the line containing $AB$ as the $x$-axis and the midpoint of $AB$ as the origin.
(I) Find the equation of the trajectory $E$ of moving point $C$;
(II) Given fixed point $P(4, 0)$, non-vertical moving line $l$ intersects trajectory $E$ at points $M$ and $N$. If lines $MP$ and $NP$ are symmetric about line $AB$, find the range of values for the area of $\triangle PMN$.

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