
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
In $\triangle ABC$, the sides $a$, $b$, and $c$ are opposite to the angles $A$, $B$, and $C$ respectively.
(1) If $\begin{vmatrix} 2c & (2a-b)\sin A \\ 1+\frac{(2b-a)\sin B}{(2a-b)\sin A} & \sin C \end{vmatrix} = 0$, determine the measure of angle $C$;
(2) If $\sin A = \frac{4}{5}$, $C = \frac{2\pi}{3}$, and $c = \sqrt{3}$, find the area of $\triangle ABC$.

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