
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 wine glass is represented by the function \( y = x^{4} \). A peach with a radius of \( r \) is placed inside the glass. What is the maximum value of the radius \( r \) such that the peach can touch the bottom of the glass (i.e., when the maximum radius \( r \) of the circle can be located in the region \( y \geqslant x^{4} \) and pass through the origin)?

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