
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 $f(x)=2\cos ^{2}x+2 \sqrt {3}\sin x\cos x$,
(1) Find the smallest positive period and the monotonically increasing interval of $f(x)$.
(2) In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. If $(a+2c)\cos B=-b\cos A$, find the range of $f(A)$.

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