
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
The cross-section of a large goblet is represented by the graph of the function \( y = x^4 \). A small sphere with radius \( r \) is placed in the goblet. Determine the maximum value of \( r \) such that the sphere can touch the lowest point of the goblet's bottom (i.e., what is the maximum value of \( r \), when the sphere with radius \( r \), located in the region \( y \geq x^4 \), can pass through the origin)?

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