
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 triangle $ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$. Let vectors $\vec{m} = (a, b)$, $\vec{n} = (\sin A, \cos B)$, and $\vec{P} = (1, 1)$.

(I) If $\vec{m}$ is parallel to $\vec{n}$, find the measure of angle $B$.

(II) If $\vec{m} \cdot \vec{P} = 4$, edge length $c = 2$, and angle $C = \frac{\pi}{3}$, find the area of triangle $ABC$.

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