
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 an acute-angled $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, satisfying $a^{2}+b^{2}=6ab\cos C$, and $\sin^{2}C=2\sqrt{3}\sin A\sin B$.

(Ⅰ) Find the value of angle $C$;

(Ⅱ) Let the function $f(x)=\sin (\omega x+\frac{\pi }{6})+\cos \omega x$ ($\omega > 0$), the distance between two adjacent highest points on the graph is $\pi$, find the range of $f(A)$.

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