
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 triangle $ABC$ with $AB = 45$, $BC = 60$, and $AC = 75$, find the area of the set of all fold points of $\triangle ABC$, assuming that all creases formed by folding $A$, $B$, and $C$ onto a fold point $P$ do not intersect inside $\triangle ABC$. The area is expressible as $m\pi-n\sqrt{t}$, where $m, n, t$ are positive integers and $t$ is not divisible by the square of any prime. Find $m+n+t$.

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