
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 a height of $300\sqrt{7}$. A fly starts at a point on the surface of the cone whose distance from the vertex is $150$, and crawls along the surface of the cone to a point on the exact opposite side whose distance from the vertex is $450\sqrt{2}$. Find the least distance that the fly could have crawled.

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