
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 $ABC$, $a$, $b$, $c$ are respectively the angles at $A$, $B$, $C$, and vectors $\overrightarrow{m} = (2\sin B, 2-\cos 2B)$, $\overrightarrow{n} = (2\sin^2\left(\frac{B}{2} + \frac{\pi}{4}\right), -1)$ and $\overrightarrow{m} \perp \overrightarrow{n}$  
(1) Find the magnitude of angle $B$;  
(2) If $a = \sqrt{3}$, $b = 1$, find the value of $c$.

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