
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 the function $f(x) = \sqrt {3}\sin \frac {x}{2}\cos \frac {x}{2} + \cos^{2} \frac {x}{2} - \frac {1}{2}$, for a triangle $\triangle ABC$ with angles $A, B,$ and $C$ and the respective opposite sides $a, b,$ and $c$:
(I) Determine the intervals where $f(x)$ is monotonically increasing.
(II) If $f(B + C) = 1$, $a = \sqrt {3}$, and $b = 1$, find the magnitude of angle $C$.

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