
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 $R$ and $S$ lie outside the plane of $\triangle DEF$ and are on opposite sides of the plane. Furthermore, $RA=RB=RC$, and $SA=SB=SC$, and the planes containing $\triangle RDE$ and $\triangle SDE$ form a $150^{\circ}$ dihedral angle. There is a point $M$ whose distance from each of $D$, $E$, $F$, $R$, and $S$ is $k$. Find $k$.

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