
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 $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 (a > b > 0)$, and the circle $O: x^2+y^2=b^2$, draw two tangents from any point $P$ on the ellipse (which does not coincide with the vertices) to the circle $O$. Let the points of tangency be $A$ and $B$, and let the line $AB$ intersect the $x$-axis and $y$-axis at points $M$ and $N$, respectively. Find the value of $\dfrac{a^2}{|ON|^2}+\dfrac{b^2}{|OM|^2}=$ \_\_\_\_\_\_\_\_\_\_\_\_\_

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