
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
In the cartesian coordinate plane $(xoy)$, the asymptotes of the hyperbola $C_{1}:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ intersect the ellipse $C_{2}:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a>b>0)$ at points $A$ and $B$ in the first and second quadrants, respectively. If the circumcircle of triangle $OAB$ has a center at $(0,\sqrt{2}a)$, find the value of $\frac{a}{b}$.

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