
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
Moor made a lopsided ice cream cone. It turned out to be an oblique circular cone with the vertex directly above the perimeter of the base (see diagram below). The height and base radius are both of length  $1$ . Compute the radius of the largest spherical scoop of ice cream that it can hold such that at least  $50\%$  of the scoop’s volume lies inside the cone.

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