
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 $\triangle ABC$ has side length $300$. Points $P$ and $Q$ lie outside the plane of $\triangle ABC$ and are on opposite sides of the plane. Furthermore, $PA=PB=PC$ and $QA=QB=QC$, and the planes of $\triangle PAB$ and $\triangle QAB$ are $200$ units away from a point $O$, which is equidistant from $A, B, C, P,$ and $Q$. Find this distance $d$.

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