
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
(1) Given vectors $\overrightarrow{a}=(2,-1)$ and $\overrightarrow{b}=(-1,m)$, if $\overrightarrow{a}\parallel\overrightarrow{b}$, then find the value of $m$.

(2) In $\triangle ABC$, where angles $A$, $B$, and $C$ are opposite to sides $a$, $b$, and $c$ respectively, if $a=1$, $c=\sqrt{3}$, and $C=\frac{\pi}{3}$, then find the value of angle $A$.

(3) Given unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a}-\overrightarrow{b}|=1$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

(4) For $\triangle ABC$, consider the following four propositions:
   $①$ If $\sin 2A = \sin 2B$, then $\triangle ABC$ is an isosceles triangle.
   $②$ If $\sin B = \cos A$, then $\triangle ABC$ is a right triangle.
   $③$ If $\sin^2 A+\sin^2 B < \sin^2 C$, then $\triangle ABC$ is an obtuse triangle.
   $④$ If $\frac{a}{\cos \frac{A}{2}} = \frac{b}{\cos \frac{B}{2}} = \frac{c}{\cos \frac{C}{2}}$, then $\triangle ABC$ is an equilateral triangle.
Identify which propositions are correct.

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