
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 $\dfrac{x^{2}}{a^{2}}+ \dfrac{y^{2}}{b^{2}}=1$ are on the $x$-axis, and a tangent line to the circle $x^{2}+y^{2}=4$ passing through point $C(2,1)$ intersects the circle at points $A$ and $B$, such that line $AB$ exactly passes through the right focus and the top vertex of the ellipse, then the equation of the ellipse is __________.

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