
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 a right triangle $DEF$ with $DE=45$, $DF=60$, and $\angle E=90^\circ$. Let us determine the area of the set of all fold points of $\triangle DEF$ that can be written as $q\pi - r\sqrt{s}$, where $q, r, s$ are positive integers and $s$ is not divisible by the square of any prime. Find $q+r+s$.

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