
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 are different, and that the stool still stands on the floor, albeit at an angle, with all four ends of the legs touching the floor. Grandpa decided to fix the stool, but he only found three pieces with lengths of 8 cm, 9 cm, and 10 cm. What could be the length of the fourth piece?

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