
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
In acute triangle $ \Delta ABC $, the sides opposite angles $A, B, C$ are denoted as $a, b, c$, respectively. Let vector $ \overrightarrow{m}=(2,c) $ and vector $ \overrightarrow{n} =\left(\frac{b}{2}\cos C-\sin A, \cos B\right) $, given that $b=\sqrt{3}$ and $ \overrightarrow{m} \bot \overrightarrow{n} $.

(Ⅰ) Find angle $B$;

(Ⅱ) Find the maximum area of $ \Delta ABC $ and the lengths of the other two sides $a, c$ under this condition.

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