
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 $\triangle ABC$, the lengths of the sides opposite to the angles A, B, and C are denoted by $a$, $b$, and $c$ respectively. It is given that $a^2-c^2=2b$ and $\sin A \cos C = 3\cos A \sin C$.  
(Ⅰ) Find the value of $b$.  
(Ⅱ) If $a=6$, find the area of $\triangle ABC$.

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