
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
If the foci of the ellipse $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1$ (where $a > b > 0$) lie on the $x$-axis, and a tangent line is drawn to the circle $x^{2}+y^{2}=1$ through the point $(1, \dfrac{1}{2})$, with the tangency points designated as $A$ and $B$, such that the line $AB$ passes exactly through the right focus of the ellipse and the top vertex, then find the equation of the ellipse.

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