
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 that $A$, $B$, and $C$ are the three internal angles of $\triangle ABC$, and vectors $\overset{⇀}{m} =(-1,\sqrt{3} )$ and $\overset{⇀}{n} =(\cos A, \sin A)$, with $\overrightarrow{m}\bullet \overrightarrow{n}=1$ and $\frac{1+\sin 2B}{{{\cos }^{2}}B-{{\sin }^{2}}B}=-3$, find $\cos C$.

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