
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
A rectangle has sides of lengths 12 and 13. An equilateral triangle is drawn such that no point of the triangle lies outside the rectangle. Find the maximum possible area of the triangle, expressed in the form $p\sqrt{q} - r$, where $p$, $q$, and $r$ are positive integers, with $q$ not divisible by the square of any prime number. Determine $p+q+r$.

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