
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 two circles $C_1: (x-1)^2+y^2=\left(\frac{7\sqrt{3}}{4}\right)^2$ and $C_2: (x+1)^2+y^2=\left(\frac{\sqrt{3}}{4}\right)^2$, a moving circle $C$ is internally tangent to $C_1$ and externally tangent to $C_2$. Let the trajectory of the center of circle $C$ be curve $G$. If a moving line $l$ intersects curve $G$ at points $P$ and $Q$, and the area of $\triangle OPQ$ is $\frac{\sqrt{6}}{2}$, where $O$ is the origin of the coordinate system.  
(Ⅰ) Find the equation of curve $G$.  
(Ⅱ) Let the midpoint of segment $PQ$ be $M$, find the maximum value of $|OM|-|PQ|$.

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