
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 $\sin \theta$ and $\cos \theta$ are two real roots of the equation $2{x^2}-(\sqrt{3}-1)x+m=0$.
$(1)$ Find the value of the real number $m$;
$(2)$ Find the value of $\frac{{\sin \theta}}{{1-\frac{1}{{\tan \theta}}}}+\frac{{\cos \theta}}{{1-\tan \theta}}$;
$(3)$ If $θ∈(\frac{3}{2}π,2π)$, find the value of $\cos 2\theta$.

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