
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
Consider a rectangular room with a square table initially placed in one of its corners. The table has dimensions \(6'\times 8'\). The owners wish to reposition the table to another corner in the manner shown, maintaining the table parallel to the walls at all times (without tilting or disassembling it). Determine the minimal integer side length \( S \) of the room required to accommodate this movement of the table.

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