
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 that $M(\sqrt{3},\frac{\sqrt{3}}{2})$ is a point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (a > b > 0)$, and the eccentricity of the ellipse is $e=\frac{1}{2}$.

(Ⅰ) Find the equation of the ellipse;

(Ⅱ) A line $m$ passing through point $P(0,3)$ intersects the ellipse at points $A$ and $B$. If $A$ is the midpoint of $PB$, find the equation of line $m$.

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