
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: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ (where $a>b>0$) with its left and right foci being $F_1$ and $F_2$ respectively, and its eccentricity is $\frac {1}{2}$. Point A is on ellipse C, $|AF_1|=2$, and $\angle F_1AF_2=60^\circ$. A line $l$ that is not perpendicular to the coordinate axes passes through $F_2$ and intersects ellipse C at points P and Q.  
(Ⅰ) Find the equation of ellipse C;  
(Ⅱ) If the midpoint of P and Q is N, is there a point M ($m$, 0) on the line segment $OF_2$ such that $MN \perp PQ$? If it exists, find the range of the real number $m$; if not, explain why.

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