
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 a rectangle $PQRS$ with sides of lengths $12$ and $5$, find the maximum possible area of an equilateral triangle such that part of every side of the triangle lies within the rectangle. Express the maximum area of such a triangle in the form $a\sqrt{b}-c$, where $a$, $b$, and $c$ are positive integers, and $b$ is not divisible by the square of any prime number. Determine $a+b+c$.

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