
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 $f(x)=2 \sqrt {3}\sin x\cos x+2\cos ^{2}x-1$.  
$(1)$ Find the maximum value of $f(x)$, and the set of values of $x$ when the function takes its maximum value;  
$(2)$ In $\triangle ABC$, $a$, $b$, and $c$ are the lengths of the sides opposite to angles $A$, $B$, and $C$ respectively, with $a=1$, $b= \sqrt {2}$, and $f(A)=2$. Find angle $C$.

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