
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
A right circular cone has a base radius of $800$ and height of $300\sqrt{3}$. A beetle starts from a point on the surface of the cone that is $150$ units away from the vertex and crawls to a point exactly opposite on the surface, at a distance of $450\sqrt{2}$ from the vertex. Determine the minimum path the beetle could have traveled on the surface of the cone.

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