
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 $$C_{1}: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a>b>0)$$ whose right focus coincides with the focus F of the parabola $C_{2}: y^{2}=4x$, and the intersection of ellipse $C_{1}$ and parabola $C_{2}$ in the first quadrant is point P, with $$|PF|= \frac {5}{3}$$.
(1) Find the equation of the ellipse $C_{1}$;
(2) A line passing through point A (-1, 0) intersects the ellipse $C_{1}$ at points M and N. Find the equation of the trajectory of the moving point R that makes $$\overrightarrow {FM}+ \overrightarrow {FN}= \overrightarrow {FR}$$ hold.

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