
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 $f(x) = \cos (2x+\frac{\pi }{3})+2\cos^2x$.

(Ⅰ) Find the maximum value of the function $f(x)$ and its smallest positive period;

(Ⅱ) Let $A$, $B$, and $C$ be the three interior angles of $\Delta ABC$. If $f(\frac{C}{2})=1$ and $c=2$, find the maximum value of the area $S$ of $\Delta ABC$.

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