
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 equation: \\( \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a > b > 0)\\), \\(F\_{1}(-1,0)\\), \\(F\_{2}(1,0)\\) are the left and right foci of the ellipse, respectively. \\(P\\) is a point on the ellipse, and \\(\\triangle PF\_{1}F\_{2}\\) has an inscribed circle \\(⊙O\_{1}\\) and a circumscribed circle \\(⊙O\_{2}\\). When \\(∠F\_{1}PF\_{2}=30^{\\circ}\\), the radius of \\(⊙O\_{1}\\) is \\(2- \\sqrt {3}\\).

1. Find the ellipse equation.
2. Let \\(S\_{2}\\) be the area of \\(⊙O\_{2}\\) and \\(S\_{1}\\) be the area of \\(⊙O\_{1}\\). Find the minimum value of \\( \frac {S\_{2}}{S\_{1}}\\).

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