
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 $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite angles $A$, $B$, and $C$, respectively. Given that $bcos(\frac{3\pi}{2}+A)+sin(\pi+B)\sqrt{\frac{6}{1-cos2C}}=0$. Find the value of $c\sin A$. If $2(b\sin C-a\tan C)=c\tan C$ and $S_{\triangle ABC}\geqslant \lambda$, determine the range of real number $\lambda$.

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