
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 $C:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with left and right foci $F_{1}(-2,0)$ and $F_{2}(2,0)$ respectively. Let $A$ be the left vertex of the ellipse $C$. The circle with diameter $F_{1}F_{2}$ intersects the ellipse $C$ in the first and second quadrants at points $M$ and $N$ respectively. If the product of the slopes of lines $AM$ and $AN$ is $\frac{1}{3}$, then the standard equation of the ellipse $C$ is ______.

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