
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 $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ with its left directrix being $$x=- \frac {3 \sqrt {2}}{2}$$, and $$a= \sqrt {3}b$$. Lines $l_{1}$ and $l_{2}$ pass through the origin $O$ with angles of 30° and 150°, respectively. Point $A$ is on line $l_{1}$, and point $B$ is on line $l_{2}$. Point $P$ satisfies $$\overrightarrow {AP}=\lambda \overrightarrow {PB}$$ ($\lambda＞0$), and point $P$ is exactly on the hyperbola $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1$$,  
(1) Find the equation of the ellipse;  
(2) Find the minimum value of the area of $\triangle OAB$.

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