
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 angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\overrightarrow{m} = (\cos B, \cos C)$ and $\overrightarrow{n} = (2a + c, b)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.

(I) Find the measure of angle $B$ and the range of $y = \sin 2A + \sin 2C$;

(II) If $b = \sqrt{13}$ and $a + c = 4$, find the area of triangle $ABC$.

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