
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 square sheet of colored paper with an area of $18 \text{ cm}^2$. One side is white and the opposite side is black. After the sheet is folded such that vertex $A$ rests exactly on the square's diagonal, creating an isosceles triangle of black area on top, the visible black area equals the visible white area. Find the distance $A$ has shifted from its original position. Present your answer as a simple radical.

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