
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
Let $x \in \mathbb{R}$, and the function $f(x) = \cos x(2 \sqrt {3}\sin x - \cos x) + \cos ^{2}\left( \frac {\pi}{2} - x\right)$.  
(I) Find the intervals of monotonic increase for the function $f(x)$ on $[0,\pi]$.  
(II) Given an acute triangle $ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, and $\frac {a^{2}+c^{2}-b^{2}}{c} = \frac {a^{2}+b^{2}-c^{2}}{2a-c}$, find the range of $f(A)$.

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