
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 $C$ with its center at the origin and its foci on the x-axis, and its eccentricity is $\frac{1}{2}$. The maximum distance from a point on the ellipse $C$ to a focus is 3.  
(Ⅰ) Find the standard equation of the ellipse $C$;  
(Ⅱ) If a line $l$ passing through point $P(0, m)$ intersects the ellipse $C$ at two distinct points $A$ and $B$, and $\overrightarrow{AP} = 3\overrightarrow{PB}$, find the range of the real number $m$.

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