
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
Greek mathematician Helen gave a formula for calculating the area of a triangle using the lengths of its three sides in her book "Measurement Theory" or "Geodesy." The formula states that for a triangle with side lengths $a$, $b$, and $c$, the area $S$ is given by $S=\sqrt{p(p-a)(p-b)(p-c)}$, where $p=\frac{1}{2}(a+b+c)$. This formula is known as the Heron's formula, which is elegant in form and reflects the symmetry of mathematics. Given that the perimeter of triangle $\triangle ABC$ is $18$ and $\sin A:\sin B:\sin C=2:3:4$, the area of $\triangle ABC$ is ______.

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