
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
In the work "Metrica" by the Greek mathematician Heron, the famous Heron's formula is recorded, which uses the lengths of the three sides of a triangle to calculate its area. If the lengths of the three sides of the triangle are $a$, $b$, and $c$, then its area $S=\sqrt{p(p-a)(p-b)(p-c)}$, where $p=\frac{1}{2}(a+b+c)$. Given that in $\triangle ABC$, $BC=6$, $AB=2AC$, find the maximum value of $\sin A$ when the area is maximized.

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