
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 its left and right foci as $F_1$ and $F_2$, and its eccentricity as $\frac{\sqrt{3}}{3}$, a line passing through $F_2$ intersects $C$ at points $A$ and $B$. Find the perimeter of $\triangle AF_1B$ in terms of $a$ and $b$.

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