
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 famous mathematician of the Southern Song Dynasty, Qin Jiushao, discovered the "three-oblique formula" for finding the area of a triangle from its three sides. In $\Delta ABC$, let the sides opposite angles $A$, $B$, and $C$ be $a$, $b$, and $c$, respectively, and the area be $S$. The "three-oblique formula" is $S=\sqrt{\frac{1}{4}\left[ {{a}^{2}}{{c}^{2}}-{{\left( \frac{{{a}^{2}}+{{c}^{2}}-{{b}^{2}}}{2} \right)}^{2}} \right]}$. If ${{a}^{2}}\sin C=4\sin A$ and ${{\left( a+c \right)}^{2}}=12+{{b}^{2}}$, use the "three-oblique formula" to find the area of $\Delta ABC$.

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