
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
In an equilateral triangle $\mathrm{ABC}$ with side length $4$, points $P$ and $Q$ are located on opposite sides of the plane of $\triangle \mathrm{ABC}$ such that $PA = PB = PC$ and $QA = QB = QC$. The dihedral angle $\angle P-AB-Q$ is $120^{\circ}$. If there exists a point $O$ equidistant from points $A$, $B$, $C$, $P$, and $Q$, find the distance $\mathrm{d}$ from $O$ to any of these points.

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