
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 three sides a, b, and c of a triangle with a < b < c, where a + b + c = 10 + 2√7, and sin(A) : sin(B) : sin(C) = 2 : √7 : 3, using the formula S = √((1/4)[c^2a^2 - (c^2 + a^2 - b^2)^2/4]/ (where a < b < c), determine the area of triangle ABC.

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