
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 $C$: $\frac{x^2}{9}-\frac{y^2}{b^2}=1$ $(b>0)$ with left and right foci $F_{1}$ and $F_{2}$, point $A$ on the left branch of $C$, $AF_{2}$ intersects the right branch of $C$ at point $B$, $\angle AF_{1}B=\frac{2\pi}{3}$, $({\overrightarrow{F_1A}+\overrightarrow{F_1B}})\cdot\overrightarrow{AB}=0$, then the focal length of $C$ is ______, and the area of $\triangle AF_{1}F_{2}$ is ______.

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