
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  $600$ . Points  $P$  and  $Q$  lie outside of the plane of  $\triangle ABC$  and are on the opposite sides of the plane. Furthermore,  $PA=PB=PC$ , and  $QA=QB=QC$ , and the planes of  $\triangle PAB$  and  $\triangle QAB$  form a  $120^{\circ}$  dihedral angle (The angle between the two planes). There is a point  $O$  whose distance from each of  $A,B,C,P$  and  $Q$  is  $d$ . Find  $d$ .

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