
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 triangle $\triangle ABC$, $c=2b\cos B$, $\angle C=\frac{2\pi}{3}$.  
$(Ⅰ)$ Find $\angle B$;  
$(Ⅱ)$ Choose one of the following three conditions as known to ensure the existence and uniqueness of $\triangle ABC$, and find the length of the median on side $BC$.  
Condition 1: $c=\sqrt{2}b$;  
Condition 2: The perimeter of $\triangle ABC$ is $4+2\sqrt{3}$;  
Condition 3: The area of $\triangle ABC$ is $\frac{3\sqrt{3}}{4}$.  
Note: If the chosen condition does not meet the requirements, no points will be awarded for question $(Ⅱ)$; if multiple suitable conditions are chosen and answered separately, the first answer will be scored.

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