
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
Triangle ABC has sides AB = 2, BC = 1, with angle C being a right angle. Take D on the segment BC, E on the segment CA, and F on the segment AB. What is the smallest possible value of the maximum length among DE, EF, and FD?

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