
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 two foci $F_1(-1, 0)$ and $F_2(1, 0)$, and the ellipse C passes through point $P\left(\frac{4}{3}, \frac{1}{3}\right)$. The equation of ellipse C is __________.

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