
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
It is known that the moving circle $P$ is tangent to the circle $F_1$: $(x+3)^2+y^2=81$ and internally tangent to the circle $F_2$: $(x-3)^2+y^2=1$. Let the trajectory of the center of circle $P$ be the curve $C$. Suppose $Q$ is a point on the curve $C$ which is not on the $x$-axis, $O$ is the origin. Through point $F_2$, construct a parallel line to $OQ$ that intersects the curve $C$ at two distinct points $M$ and $N$.

(Ⅰ) Find the equation of curve $C$.

(Ⅱ) Investigate whether the ratio $|MN|$ to $|OQ|^2$ can be a constant. If so, find this constant; if not, explain why.

(Ⅲ) Let the area of $\triangle QF_2M$ be $S_1$ and the area of $\triangle OF_2N$ be $S_2$. Let $S=S_1+S_2$, find the maximum value of $S$.

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