
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
Let the quadrilateral formed by connecting the four vertices of the hyperbolas $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1$$ and $$\frac {y^{2}}{b^{2}}- \frac {x^{2}}{a^{2}}=1$$ $(a>0, b>0)$ have an area of $S_1$, and the quadrilateral formed by connecting their four foci have an area of $S_2$. Then, the maximum value of $$\frac {S_{1}}{S_{2}}$$ is \_\_\_\_\_\_.

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