
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 the ellipse $C:\dfrac{x^2}{4}+y^2=1$.

(I) If $P$, $Q$ are two points on the ellipse $C$, and the midpoint of the segment $PQ$ is $\left( 1,\dfrac{1}{2} \right)$, find the slope of the line $PQ$;

(II) For a line passing through $\left(1,0\right)$ and not perpendicular to the x-axis, intersecting the ellipse at points $M$, $N$, is there a fixed point $A$ on the x-axis such that $\angle OAM=\angle OAN$? If it exists, find the coordinates of the fixed point $A$; if not, explain why.

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