
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 the hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $(a>0, b>0)$ with left and right foci $F_{1}$ and $F_{2}$, eccentricity $\frac{2\sqrt{3}}{3}$, and the asymptote line $y=\frac{b}{a}x$ passing through $F_{2}$, intersects $C$ at points $A$ and $B$ when a line perpendicular to $y=\frac{b}{a}x$ passing through $F_{2}$, where point $A$ is in the first quadrant. If $|AF_{2}|=\frac{3-\sqrt{3}}{2}$, then the perimeter of $\triangle ABF_{1}$ is ____.

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