
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 opposite to the angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Define vectors $\overrightarrow{m}=(\sin B + \sin C, \sin A + \sin B)$ and $\overrightarrow{n}=(\sin B - \sin C, \sin A)$. It is given that $\overrightarrow{m} \perp \overrightarrow{n}$.

$(1)$ Find the magnitude of angle $C$.

$(2)$ If $c= \sqrt{3}$, determine the range of possible values for $2a+b$.

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