
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 circle $x^2+y^2=9$, point $A(1,1)$ is inside the circle, and $P$, $Q$ are moving points on the circle such that $\angle PAQ=90^\circ$, $M$ is the midpoint of $PQ$.
(1) Find the equation of the trajectory curve $C$ of point $M$.
(2) Let $E(\frac{9}{2}, \frac{1}{2})$, $D(\frac{1}{2}, \frac{1}{2})$, determine whether there exists a point $F$ on the line $ED$ and distinct from point $E$, such that the ratio $\frac{|HE|}{|HF|}$ is constant for any point $H$ on curve $C$. If exists, find the coordinates of point $F$; if not, explain the reason.

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