
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 $Q$ are outside the plane of $\triangle ABC$, one on each side of the plane. Further, $PA = PB = PC = 350$ and $QA = QB = QC = 350$. The planes of $\triangle PAB$ and $\triangle QAB$ form a $60^\circ$ dihedral angle. Find the length of segment $d$, where $d$ is the distance from a point $O$ to $A, B, C, P$, and $Q$, with $O$ located equidistantly from these points.

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