
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 the ellipse C: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a>b>0)$$, where F<sub>1</sub> and F<sub>2</sub> are the left and right foci, respectively, and the eccentricity is $$\frac {1}{2}$$. Point A is on ellipse C, $$| \overrightarrow {AF_{1}}|=2$$, $$| \overrightarrow {AF_{2}}|| \overrightarrow {F_{1}A}|=-2 \overrightarrow {AF_{2}}\cdot \overrightarrow {F_{1}A}$$. A line l, which is not perpendicular to the coordinate axes, passes through F<sub>2</sub> and intersects the ellipse at points P and Q.  
(1) Find the equation of ellipse C;  
(2) Does there exist a point M(m, 0) on the line segment OF<sub>2</sub> such that the quadrilateral with sides MP and MQ as adjacent sides is a rhombus? 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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