
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 following conditions:①$\left(a+b+c\right)\left(a+b-c\right)=3ab$, ②$\frac{{tanA+tanB}}{{tanAtanB-1}}=\sqrt{3}$, ③$\frac{{sinC}}{{2sinB-sinA}}=\frac{{cosC}}{{cosA}}$, choose one of these three conditions and fill in the blank below, then solve the following:<br/>In $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, and satisfy __________.<br/>$(1)$ Find the value of $\tan C$;<br/>$(2)$ If $D$ is a point on side $BC$ such that $AD=6$, $BD=4$, and $AB=8$, find $AC$.

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