
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 that $c \sin A = \sqrt{3}a \cos C$ and $(a-c)(a+c)=b(b-c)$, find the period and the monotonically increasing interval of the function $f(x) = 2 \sin x \cos (\frac{\pi}{2} - x) - \sqrt{3} \sin (\pi + x) \cos x + \sin (\frac{\pi}{2} + x) \cos x$. Also, find the value of $f(B)$.

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