
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
Given a triangle $\triangle ABC$ with a perimeter of $10 + 2\sqrt{7}$ and $\sin A : \sin B : \sin C = 2 : \sqrt{7} : 3$, use the formula $S = \sqrt{\frac{1}{4}\left[c^{2}a^{2} - \left(\frac{c^{2} + a^{2} - b^{2}}{2}\right)^{2}\right]}$ to find the area of $\triangle ABC$.

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