
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
Equilateral $\triangle DEF$ has side length $300$. Points $X$ and $Y$ lie outside the plane of $\triangle DEF$ and are on opposite sides of the plane. Furthermore, $XA=XB=XC$, and $YA=YB=YC$, and the planes of $\triangle XDE$ and $\triangle YDE$ form a $90^{\circ}$ dihedral angle (the angle between the two planes). There is a point $R$ whose distance from each of $D, E, F, X,$ and $Y$ is $r$. Find $r$.

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