
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 Chinese mathematician Qin Jiushao of the Song Dynasty once proposed the "method of three oblique lines for finding the area," which assumes a triangle in a plane with side lengths $a$, $b$, and $c$. The area $S$ of the triangle can be calculated using the formula $S=\sqrt{p(p-a)(p-b)(p-c)}$, where $p$ is half of the triangle's perimeter. This formula is also known as the Heron's formula. Given a triangle with side lengths satisfying $a+b=10$, $c=6$, the maximum value of the area of this triangle is ______.

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