
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 triangle $\triangle ABC$, the lengths of the sides opposite the three interior angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If in the triangle, $\cos A = \frac{1}{3}$, $S = 4\sqrt{2}$, and $\sin \left(A-B\right) = 2\sin B\left(1-2\cos A\right)$, calculate the length of side $c$.

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