
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
The area of $\triangle ABC$ can be calculated using the "Three-Side Formula" $S= \sqrt { \frac {1}{4}[a^{2}c^{2}-( \frac {a^{2}+c^{2}-b^{2}}{2})^{2}]}$. Given that $a^{2}\sin C=4\sin A$ and $(a+c)^{2}=12+b^{2}$, calculate the area of $\triangle ABC$.

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