
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 the function $f(x)=(2\sqrt{3}\sin x-\cos x)\cos x+\cos^2\left(\frac{\pi}{2}-x\right)$.

$(1)$ Find the intervals of monotonic increase for the function $f(x)$ on the interval $[0,\pi]$;

$(2)$ Let $\triangle ABC$ be an acute triangle with internal angles $A$, $B$, $C$ and opposite sides $a$, $b$, $c$ respectively, and $\frac{a^2+c^2-b^2}{c}=\frac{a^2+b^2-c^2}{2a-c}$. Find the range of values for $f(A)$.

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