
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
Given a thin, square plate made of a homogeneous material with $20 \mathrm{~cm}$ sides, find the point on its perimeter where, when the plate is suspended, the nearest vertex will be the farthest from the suspension point. Determine the distance between the suspension point and the nearest vertex to a precision of one-tenth of a millimeter.

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