
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 the area of triangle $\triangle ABC$ is given by $S=\sqrt{\frac{1}{4}[{{a^2}{c^2}-{{({\frac{{{a^2}+{c^2}-{b^2}}}{2}})}^2}}]}$, $c=6$, $a+b=10$, and $a$ and $b$ are nonnegative, find the maximum value of the area of $\triangle ABC$.

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