
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 of radius $500$ and a height of $300$. A bug starts at a point on the surface of the cone whose shortest distance from the vertex is $100$, and moves along the surface to a point whose shortest distance from the vertex is $400$. Find the minimum distance that the bug must have crawled if it ended $90^\circ$ clockwise from its starting position on the cone.

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