
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 an ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a > b > 0\right)$ with eccentricity $\frac{1}{3}$, $A_{1}$ and $A_{2}$ are the left and right vertices of $C$, and $B$ is the top vertex of $C$. If $\overrightarrow{BA_{1}}\cdot \overrightarrow{BA_{2}}=-1$, determine the equation of $C$.

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