
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
Little Petya sawed off all the legs of a square stool and lost four of the cut-off pieces. It turned out that the lengths of all the pieces were different, and the stool still stands on the floor, albeit inclined, but still touching the floor with all four ends of the legs. Grandpa decided to fix the stool but found only three pieces with lengths 8, 9, and 10 cm. What is the possible length of the fourth piece?

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