
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 $PQRS$ has sides of lengths $13$ and $14$. An equilateral triangle is drawn such that no point of the triangle lies outside of $PQRS$. Find the maximum possible area of such a triangle. Express the area 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. What is $a+b+c$?

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