
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 circle $O$ is given by $x^{2}+y^{2}=4$, and a fixed point $F(1,0)$. Let $P$ be a moving point on the plane such that the circle with diameter $FP$ is tangent to circle $O$. The trace of point $P$ forms curve $C$.
(I) Find the equation of curve $C$.
(II) Let $M$ and $N$ be points on curve $C$ and the line $MN$ passes through the fixed point $(0, \frac {1}{2})$. Determine if there exists a fixed point $Q$ on the $y$-axis such that $∠MQO=∠NQO$. If such a point exists, find its coordinates; if not, provide an explanation.

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