
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 an ellipse $C:\frac{{{x^2}}}{{{a^2}}}+\frac{{{y^2}}}{{{b^2}}}=1 (a>b>0)$ with eccentricity $\frac{1}{2}$, point $P(1,\frac{3}{2})$ lies on the ellipse, and points A, B are two distinct points on the ellipse, with O as the origin. <br /> (Ⅰ) Find the equation of ellipse C; <br /> (Ⅱ) Let M be the midpoint of line segment AB. When the area of triangle AOB is maximized, are there two fixed points G, H such that |GM|+|HM| is a constant value? If yes, find this constant value; if not, explain the reason.

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