
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 Qin Jiushao of the Southern Song Dynasty in China discovered a formula for calculating the area of a triangle from its three sides, which he called "San Xie Qiu Ji" (literally "area calculation with three sides"). This method filled a gap in traditional Chinese mathematics. If this method is written as a formula, it is $S=\sqrt{\frac{1}{4}[{{c^2}{a^2}-{{({\frac{{{c^2}+{a^2}-{b^2}}}{2}})}^2}}]}$, where $a$, $b$, and $c$ are the three sides of the triangle, and $S$ is the area of the triangle. Given a triangle with side lengths $a=\sqrt{2}$, $b=\sqrt{3}$, and $c=2$, the area of this triangle $S=$____.

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