
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 that $O$ is the coordinate origin, and there are two points $A$ and $B$ on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ such that $OA \perp OB$, it follows that
$$
\frac{1}{|OA|^{2}}+\frac{1}{|OB|^{2}}=\frac{1}{a^{2}}+\frac{1}{b^{2}},
$$
and the line $AB$ is tangent to the circle $x^{2}+y^{2}=\frac{a^{2}b^{2}}{a^{2}+b^{2}}$.

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