
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 foci named F<sub>1</sub> and F<sub>2</sub>, and one end point of its minor axis being point P. The radius of the inscribed circle of the triangle ΔPF<sub>1</sub>F<sub>2</sub> is $$\frac {b}{3}$$. Assuming a line l passing through F<sub>2</sub> intersects the ellipse C at points R and S, when l is perpendicular to the x-axis, |RS| equals 3.
(1) Find the standard equation of the ellipse C.
(2) Does there exist a point T on the x-axis such that, as line l changes, the line containing TS and TR is always symmetrical about the x-axis? If such a point exists, find the coordinates of T. If it does not, explain why.

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