
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 foci $F_1$ and $F_2$, and eccentricity $\frac{\sqrt{3}}{3}$. A line passing through $F_2$ intersects $C$ at points $A$ and $B$. If the perimeter of $\triangle AF_1B$ is $4\sqrt{3}$, then the equation of $C$ is \_\_\_\_\_\_, and at this time, a chord of ellipse $C$ is bisected by $(1,1)$. The equation of the line containing this chord is \_\_\_\_\_\_.

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