
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 point $P$ on a triangular piece of paper $ABC$, consider the creases formed when $A, B,$ and $C$ are folded onto $P.$ Let $P$ be a fold point of $\triangle ABC$ if these creases, which number three unless $P$ is one of the vertices, do not intersect. Suppose that $AB=45, AC=45,\,$ and $\angle C=90^\circ.\,$ Then calculate the area of the set of all fold points of $\triangle ABC.$

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