
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 hyperbola $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1(a＞0，b＞0)$$ with its two foci being F<sub>1</sub> and F<sub>2</sub>, respectively, and P is a point on its left branch. The distance from P to the left directrix is d.  
(1) If y= $$\sqrt {3}$$x is one of the asymptotes of the given hyperbola, is there a point P such that d, |PF<sub>1</sub>|, |PF<sub>2</sub>| form a geometric sequence? If such a point exists, write down the coordinates of P; if not, explain why.  
(2) On the left branch of the given hyperbola, when there exists a point P such that d, |PF<sub>1</sub>|, |PF<sub>2</sub>| form a geometric sequence, find the range of the eccentricity e.

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